universitÉdemontrÉal ... · method to allow for even more ... a point implicit point-jacobi ......

UNIVERSITÉ DE MONTRÉAL

DEVELOPMENT OF AN OVERSET STRUCTURED 2D RANS/URANSNAVIER-STOKES SOLVER USING AN IMPLICIT SPACE AND NON-LINEAR

FREQUENCY DOMAIN TIME OPERATORS

ANTOINE TAILLON LÉVESQUEDÉPARTEMENT DE GÉNIE MÉCANIQUEÉCOLE POLYTECHNIQUE DE MONTRÉAL

MÉMOIRE PRÉSENTÉ EN VUE DE L’OBTENTION DU DIPLÔME DE MAÎTRISE ÈS SCIENCES APPLIQUÉES

(GÉNIE AÉROSPATIAL)AVRIL 2015

c© Antoine Taillon Lévesque, 2015.

UNIVERSITÉ DE MONTRÉAL

ÉCOLE POLYTECHNIQUE DE MONTRÉAL

Ce mémoire intitulé :

DEVELOPMENT OF AN OVERSET STRUCTURED 2D RANS/URANSNAVIER-STOKES SOLVER USING AN IMPLICIT SPACE AND NON-LINEAR

FREQUENCY DOMAIN TIME OPERATORS

présenté par : LÉVESQUE Antoine Taillonen vue de l’obtention du diplôme de: Maîtrise ès sciences appliquéesa été dûment accepté par le jury d’examen constitué de :

M. PELLETIER Dominique, Ph. D., présidentM. LAURENDEAU Éric, Ph. D., membre et directeur de rechercheM. YANG Hong, Ph. D., membre

iii

ACKNOWLEDGMENTS

I would like to thank most of all my research supervisor, professor Eric Laurendeau. Hisinvaluable guidance, thrust and knowledge allowed me to learn so much during these twoyears.

I would also like to thank Bombardier Aerospace’s Advanced Aerodynamics group for all thein kind support provided.

This project would not have been possible without the financial support of the NaturalSciences and Engineering Research Council of Canada (NSERC), the Fond de RechercheQuébécois de Nature et Technologie (FRQNT) and Bombardier Aerospace.

To all my collegues, Thibaut Deloze and Ali Mosahebi for the support and the knowledge Ihave acquired through your teachings, Alexandre Pigeon and Simon Bourgault-Côté for thecountless hours of debbuging and code writting we have made together and to all others forthe many discussions and insights that cleared so many roadblocks.

To Sophie and my family, for their unconditional support, interest, motivation, comfort andlove.

iv

RÉSUMÉ

Ce projet s’intéresse au développement de méthodes avancées afin de réaliser des simulationsnumériques en mécanique des fluides. Plus particulièrement, ces méthodes s’appliquerontaux modèles RANS et URANS sur des profils aérodynamiques simples et multi-éléments.Ces développements seront utilisés afin de réaliser une optimisation sur l’interstice et lechevauchement d’un volet de bord de fuite ainsi que des simulations instationnaires standard.

Le logiciel utilisé comme plateforme de développement est NSCODE, un solveur Navier-Stokes bidimensionnel pour maillages structurés. Les développements logiciels seront réalisésdans un cadre rigoureux et en utilisant des techniques de programmations appropriées. Lesméthodes implémentées viseront plusieurs aspects du solveur incluant les capacités topologiques,l’opérateur spatial et l’opérateur temporel. Afin de traiter les profils multi-éléments, la méth-ode multi-blocs sera implémentée afin de partitionner le domaine de calcul. La méthodechimère est ensuite implémentée afin d’améliorer la flexibilité de la méthode multi-blocs. Leschéma de dissipation artificielle scalaire est ensuite remplacé par le schéma de dissipationmatricielle afin d’améliorer la précision du solveur. Un schéma d’opérateur spatial utilisantun préconditionneur Jacobien implicite par point ainsi qu’un schéma implicite Block Lower-Upper Symmetric Gauss Seidel (LU-SGS) sont ensuite implémentés afin d’améliorer le tauxde convergence du solveur. L’opérateur temporel par pas de temps double présent dans lelogiciel initial est adapté afin d’être compatible avec les différents schémas d’opérateurs spa-tiaux utilisés. Un opérateur temporel Non-Linéaire dans le Domaine Fréquentiel (NLFD) estensuite implémenté afin de résoudre efficacement les écoulements instationnaires périodiques.

Chacune des méthodes implémentées est validée et vérifiée en utilisant des cas tests utilisésdans la littérature ainsi qu’avec des résultats expérimentaux. Une grande variété de cas testssont utilisés afin de s’assurer de la fiabilité du solveur lors des applications futures.

Les implémentations logicielles sont ensuite utilisés afin de résoudre deux problèmes:

• Une optimisation de dispositif hypersustentateur;

• Une simulation dans le domaine fréquentiel d’un profil aérodynamique en tangage dansun écoulement turbulent.

L’optimisation du dispositif hypersustentateur vise à maximiser le coefficient de portance duprofil de recherche MDA en modifiant la position du volet de bord de fuite. En utilisantune optimisation utilisant des simulations purement bidimensionnelles en parallèle à une

v

optimisation utilisant des simulations tridimensionnelles utilisant l’hypothèse d’aile en flècheinfinie, la démonstration est faite qu’une optimisation bidimensionnelle n’est pas adaptée audesign de dispositifs hypersustentateurs sur des ailes en flèches. La seconde application apour but d’introduire un modèle de turbulence aux simulations NLFD dans NSCODE. Entant qu’étape vers l’utilisation de modèles de turbulence à une et deux équations, un modèlealgébrique est utilisé. Ce problème vérifiera donc l’utilisation d’un modèle de turbulencealgébrique sur un opérateur NLFD. La simulation NLFD d’un profil en tangage dans unécoulement turbulent donne des résultats en accord avec la littérature et avec les simulationspar pas de temps double ce qui ouvre la voie à l’utilisation de modèles de turbulence pluscomplexes avec la méthode NLFD.

vi

ABSTRACT

This project aims at performing 2D RANS and URANS computational fluid dynamics simu-lations over single and multi-element airfoil. The application of such developments is demon-strated via flap gap/overlap optimisation and standard URANS cases.

The work presented in this thesis was implemented in NSCODE, a 2D structured gridReynolds-Averaged Navier-Stokes flow solver, and is included in a solid framework to ensureits quality and maintainability. It covers many aspects of the flow solver, including topologycapabilities, steady and unsteady solver schemes. To simulate flows around complex geome-tries, the multi block technique is implemented in order to partition the computationnaldomain. The multi block capability is then expanded to overset meshes with the Chimeramethod to allow for even more flexibility in geometry treatment. The existing scalar dissi-pation scheme is replaced by the matricial artificial dissipation scheme (MATD) to increasespatial resolution accuracy. A point implicit Point-Jacobi Preconditioner and an implicitBlock Lower-Upper Symmetric Gauss Seidel (LU-SGS) space solving scheme are then imple-mented to increase convergence rates. The time discretization schemes are also improved.The baseline dual time stepping scheme is modified to be compatible with the LU-SGS solverschemes. A Non-Linear Frequency Domain is also added to the software in order to efficientlysolve periodic problems.

Each of these techniques is validated and verified against literature data and experimentaldata. A wide range of test case is chosen in order to ensure full confidence in the developpedsoftware.

The software capability developments are then used to solve two problems:

• A high-lift airfoil optimisation;

• An unsteady simulation of turbulent flows in the frequency domain of a pitching airfoil.

The high-lift airfoil optimisation seeks to maximise the lift coefficient of the McDonnel Dou-glas Research airfoil by changing the flap’s position. Using a two dimensional approachin parallel to a three dimensional approach with infinite swept wing hypothesis, a physicalphenomenon that couldn’t be previously observed on two dimensional solvers was captured.The second case sought to introduce a turbulence component to the NLFD implementationin NSCODE. As a step before using one and two equations turbulence models, an algebraicturbulence model is used in the study. This problem will thus test the applicability of an alge-braic turbulence model to the NLFD method. The NLFD resolution of a turbulent pitching

vii

airfoil yielded results that validated very well with the literature and equivalent Dual TimeStepping resolutions, paving the way for the use of more complex turbulence models in NLFDresolutions.

viii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

RÉSUMÉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF ACRONYMS AND ABREVIATIONS . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Finite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Unsteady simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Elements of the Problematics . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Computational Costs of Navier-Stokes Simulations . . . . . . . . . . . 51.2.2 Accurate Solutions of numerical Navier-Stokes Simulation . . . . . . . 51.2.3 Complex Geometry Treatment . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Plan of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

CHAPTER 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Steady Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Scalar Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Matrix Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Runge-Kutta Time Integration . . . . . . . . . . . . . . . . . . . . . . 122.1.4 Implicit Residual smoothing . . . . . . . . . . . . . . . . . . . . . . . 122.1.5 Block lower-upper symmetric Gauss-Seidel scheme . . . . . . . . . . . 13

ix

2.1.6 Multigrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Unsteady Reynolds-Averaged Navier-Stokes solvers . . . . . . . . . . . . . . 14

2.2.1 Dual Time-Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Non-Linear Frequency Domain . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Framework Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

CHAPTER 3 SOFTWARE DEVELOPMENTS . . . . . . . . . . . . . . . . . . . . 183.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Topology Capabilities Improvements . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Multiblock Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Overset mesh compatibility . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Steady Solver Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 Matrix dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Point Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.3 LU-SGS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.4 Validation and Verification . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Unsteady solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Dual Time-Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.2 Non Linear Frequency Domain Solver . . . . . . . . . . . . . . . . . . 413.4.3 Arbitrary Lagrangian Eulerian formulation . . . . . . . . . . . . . . . 433.4.4 Validation and Verification . . . . . . . . . . . . . . . . . . . . . . . . 44

CHAPTER 4 NUMERICAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . 534.1 High-lift airfoil optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 2.5D infinite swept wing . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Optimisation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Turbulent NLFD case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.1 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Pitching NACA64A010 airfoil . . . . . . . . . . . . . . . . . . . . . . 57

CHAPTER 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1 Synthesis of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Limitations of the Proposed Solution . . . . . . . . . . . . . . . . . . . . . . 615.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

x

LIST OF TABLES

Table 3.1 Euler NACA0012 allowable CFL number/ω . . . . . . . . . . . . . . 30Table 3.2 Euler NACA0012 lift and drag coefficients . . . . . . . . . . . . . . . 30Table 3.3 Convergence order and continuum estimates for subsonic and transonic

NACA0012 Euler solutions . . . . . . . . . . . . . . . . . . . . . . . . 34Table 3.4 RAE2822 allowable CFL number/ω of NSCODE compared to Cagnone

et al. (2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Table 3.5 RAE2822 lift and drag coefficients . . . . . . . . . . . . . . . . . . . . 35Table 3.6 NLR7301 allowable CFL number/ω of NSCODE compared to Cagnone

et al. (2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Table 3.7 NLR7301 lift and drag coefficients at 13.1 angle of attack . . . . . . 40Table 3.8 NLR7301 lift and drag coefficients at 5 angle of attack . . . . . . . . 40Table 3.9 CPU time to achieve 500 multigrid cycles for different number of modes

used in the NLFD resolution for the Euler pitching NACA0012 case . 45Table 3.10 Convergence order and continuum estimates for pitching NACA0012

Dual Time-Stepping solutions . . . . . . . . . . . . . . . . . . . . . . 47Table 3.11 Strouhal numbers obtained for the cylinder under laminar flow conditions 48

xi

LIST OF FIGURES

Figure 3.1 Flow chart of steady NSCODE execution . . . . . . . . . . . . . . . . 19Figure 3.2 NACA0012 overset mesh for testing of multigrid technique on chimera

grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.3 Convergence of the NACA0012 airfoil test case with overset mesh . . 23Figure 3.4 Staggered NACA0012 pressure isobars with grid M = 0.7 . . . . . . . 24Figure 3.5 Staggered NACA0012 pressure coefficients M = 0.7 . . . . . . . . . . 25Figure 3.6 Chimera and one-to-one grids used with NSCODE to compute the

McDonnell Douglas 30P30N Airfoil test case . . . . . . . . . . . . . . 26Figure 3.7 Convergence and pressure distribution obtained on the 30P30N airfoil 26Figure 3.8 Node treatment order of the LU-SGS upward sweeps . . . . . . . . . 29Figure 3.9 Residual convergence of the Euler NACA0012 test case with respect to

iterations and CPUtime . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 3.10 Mach contours of the Euler NACA0012 test case . . . . . . . . . . . . 32Figure 3.11 Pressure distribution of the Euler NACA0012 test case . . . . . . . . 32Figure 3.12 Parallel speedup factor of NSCODE . . . . . . . . . . . . . . . . . . . 33Figure 3.13 Convergence order for subsonic and transonic NACA0012 Euler solutions 34Figure 3.14 RAE2822 grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.15 Residual convergence of the RAE2822 test case with respect to itera-

tions and CPUtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 3.16 Mach contours of the RAE2822 test case . . . . . . . . . . . . . . . . 37Figure 3.17 Pressure distribution of the RAE2822 test case . . . . . . . . . . . . . 37Figure 3.18 NLR7301 grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.19 Mach contours and pressure distribution of the NLR7301 test case at

13.1 angle of attack . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.20 Residual convergence of the NLR7301 test case with respect to itera-

tions and CPUtime at 13.1 angle of attack . . . . . . . . . . . . . . . 39Figure 3.21 Residual convergence of the NLR7301 test case with respect to itera-

tions and CPUtime at 5 angle of attack . . . . . . . . . . . . . . . . 40Figure 3.22 Flow chart of unsteady NSCODE Dual Time Stepping execution . . . 42Figure 3.23 Flow chart of NLFD iteration algorithm on the state variables . . . . 43Figure 3.24 Euler pitching NACA0012 convergence characteristics versus multigrid

cycles for different number of modes . . . . . . . . . . . . . . . . . . 44

xii

Figure 3.25 Euler pitching NACA0012 convergence characteristics versus CPU timefor different number of modes . . . . . . . . . . . . . . . . . . . . . . 45

Figure 3.26 Lift and drag hysteresis of the lift and drag coefficients versus theinstantanous angle of attack . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 3.27 Order of time convergence of NSCODE using the L2 norm integral oflift and drag coefficient over a period for a pitching NACA0012 airfoil 47

Figure 3.28 Mesh used for the circular cylinder simulations . . . . . . . . . . . . . 49Figure 3.29 Vorticity contours on the circular cylinder simulations . . . . . . . . . 49Figure 3.30 Lift and drag coefficient variations over a period . . . . . . . . . . . . 50Figure 3.31 Overset mesh used on the circular cylinder simulation . . . . . . . . . 51Figure 3.32 Vorticity contours on the circular cylinder simulation using an overset

mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 3.33 Convergence of each harmonic mode on a 4 modes (top) and 10 modes

(bottom) NLFD resolution . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 4.1 Gap and overlap definition for a flap . . . . . . . . . . . . . . . . . . 54Figure 4.2 Cl–α curve of the MDA airfoil for a purely 2D simulation (0 sweep)

and an infinite swept wing simulation with a sweep of 30 . . . . . . . 55Figure 4.3 Streamlines and pressure field over a MDA geometry for a 2D (or in-

finite 0 swept wing) on the left and an infinite 30 swept wing on theright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Figure 4.4 Lift coefficient obtained for each x and y displacement of the flap in a2D flow on the left and an infinite 30 swept wing on the right . . . . 56

Figure 4.5 Mesh used for the turbulent pitching NACA64A010 test case . . . . . 57Figure 4.6 Lift and moment coefficients versus the angle of attack for the pitching

turbulent NACA64A010 airfoil . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.7 Mach contours at 0.65 in a nose down motion of the turbulent NACA64A010

airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.8 Cp distribution at 0.65 in a nose down motion of the turbulent NACA64A010

airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xiii

LIST OF ACRONYMS AND ABREVIATIONS

ALE Arbitrary Lagrangian EulerianCFD Computational Fluid DynamicsCFL Courant-Friedrichs-Lewy numberDTS Dual Time SteppingFFT Fast Fourrier TransformFFTW Fastest Fourrier Transform in the WestGCL Geometric Conservative LawIRS Implicit Residual SmoothingMATD Matrix Artificial DissipationMDA McDonnell Douglas research AirfoilNLFD Non-Linear Frequency DomainNS Navier-StokesPJ Point JacobiRANS Reynolds Averaged Navier-StokesTSL Thin Shear LayerURANS Unsteady Reynolds Averaged Navier-Stokes

xiv

LIST OF APPENDICES

Appendix A TOPOLOGY FILE EXAMPLE . . . . . . . . . . . . . . . . . . . . . 68

Appendix B VISCOUS JACOBIANS . . . . . . . . . . . . . . . . . . . . . . . . . 69

1

CHAPTER 1 INTRODUCTION

For more than fifty years, the rapid development of computational fluid dynamics (CFD)has completely changed the way aircrafts are designed. The advancements in fluid dynam-ics physics understanding and the availability of exponentially growing computing powerallowed CFD to become a major analysis tool for aerodynamic design. More and more, CFDsimulations tend to be used alongside wind tunnel testing due to their increased accuracyand reduced costs. Although numerical simulations via Reynolds-Averaged Navier-Stokes(RANS) solvers are regarded as adequate tools for cruise design conditions, further develop-ments are still required to accurately and efficiently compute flows over high-lift devices aswell as unsteady flows. New methods are still being developed and investigated to resolvethese issues in order to widen the use of CFD in the aircraft design process.

1.1 Basic Concepts

1.1.1 Navier-Stokes Equations

The CFD methods examined in this work are solving the Reynolds-Averaged Navier-Stokesequations. These equations consists of transport equations solving conservation of mass,momentum and energy in the fluid. The general formulation of these equations in threedimensions is (Versteeg and Malalasekera, 2007)

∂ρ

∂t+ div (ρ~u) = 0

∂

∂t(ρu) + div (ρ~uu) = −∂p

∂x+ div (µ grad(u)) + SMx

∂

∂t(ρv) + div (ρ~uv) = −∂p

∂y+ div (µ grad(v)) + SMy

∂

∂t(ρw) + div (ρ~uw) = −∂p

∂x+ div (µ grad(w)) + SMz

∂

∂t(ρE) + div (ρ~uE) = −pdiv(~u) + div (k grad(T )) + Φ + SE (1.1)

where µ the viscosity coefficient and k the heat conductivity are fluid properties, ρ is thedensity, u, v, w are the cartesian components of the fluid velocity ~u. The source terms Srepresent other forces and energy sources acting on the fluid such as plasma actuators orgravity.

2

Equations 1.1 can be rewritten as a general transport equation using φ as a general conser-vative variable

∂ (ρφ)∂t

+ div (ρ~uφ) = div (Γ grad (φ)) + Sφ (1.2)

with Γ the appropriate fluid property depending on the equation. Equation 1.2 is solvednumerically using space and time discretisation operators.

1.1.2 Finite Volume

In CFD applications, three main space discretisation techniques are used to solve equations1.2: the finite volume method, the finite difference method and the finite element method.The finite volume method is the most widely used on modern CFD software within theaeronautical industry due to its ability to naturally model transport equations.

The principle of the method is to integrate equation 1.2 on a control volume to yield

∂

∂t

∫CV

ρφdCV = −∫A~n · (ρ~uφ) dA+

∫A~n · (Γ grad(φ)) dA+

∫CV

SφdV (1.3)

Where CV represent the control volume and A the boundaries of the volumes. A physicalinterpretation of equation 1.3 is that the rate of change of the variable φ in the control volumeis equal to the flux of φ passing through the boundaries of the control volume, plus thediffusion of φ through the boundaries, and the source of φ within the volume. Discretizationin time determines the physical time step ∆t. In the case where the steady state solutionis sought, a pseudo-time step is defined based on stability bounds instead of the physics ofthe problem. The right hand side of equation 1.3 is often called residual and is treated asconvective residual

∫A n · (ρ~uφ) dA, viscous residual

∫A n · (Γ grad(φ)) dA and source terms

residuals∫CV SφdV . Convergence is obtained when the solution leads to a residual of zero.

1.1.3 Meshing

Often undervalued in flow simulations, the mesh is a very important component of any CFDmethods. Mesh quality is critical to capture the desired physical phenomena as well asobtaining fast solution convergence. The role of the mesh, also called grid, is to cover all thecomputational domain with nodes on which the space discretisation applies. For the purposeof this work, only conformal meshes will be treated. Those are the meshes whose boundariescoincide with the computational domain boundaries.

Meshes are classified into two main categories, structured and unstructured grid. Structuredgrids are Cartesian grid in the topological space transformed to conform to a specific domain.

3

In a structured grid, each node can be identified via its topological coordinates i, j, k forthree dimensional meshes or i, j for two dimensional meshes. This property allows for mucheasier treatment (Blazek, 2005) since the neighbours of each node can be found quickly.Unstructured grids are not arranged in this manner in the topological space and additionalinformation must be stored to find the neighbours of each nodes. The constraint of organisingthe grid in topological space however leads to difficulties treating complex domains. Inexternal aerodynamics, many simulations have complex geometries involving more than onebody such as wing-aileron configuration, wing-flap configuration, wing-fuselage configuration,etc.

Structured meshes are divided into two main sub-categories, single block and multiblock grids.Single block grids are the simplest form of structured meshes and are simply a Cartesian gridin the topological domain deformed to adapt to a specific computational domain. Multiblockmeshes can be seen as an assembly of several blocks where the domain is subdivided intosections, each of the section being covered with a different single block mesh. There aredifferent types of multiblock meshes depending on the constraints on block boundaries. Oneof the most used multiblock mesh type is the one to one where different blocks have thesame points distributions along their common boundaries, providing continuity in the meshat the block boundaries. This property of one to one meshes allow for easier connectiontreatment between blocks. Sliding meshes are similar to one to one except that the nodedistribution on either side of boundaries are not identical leading to additional work to ensureflow conservation at the block boundaries.

A very interesting multiblock mesh type is the overset meshes, also called chimera meshes.The main difference of overset meshes compared to other multiblock grids is that the blocksare not required to be adjacent to one another. Indeed, blocks can overlap each other arbi-trarily in this type of meshes, removing topological constraints during grid creation comparedto multiblock meshes. Communication between overset blocks are normally assured via in-terpolations and involve more work than structured multiblock meshes. The computationrequired to set up chimera meshes for the solver are typically done in a dedicated preprocessor(Suhs et al., 2002).

1.1.4 Unsteady simulations

Unsteady simulations are required at many stages in aircraft design. Gas turbine, rotor craftflights and aircraft stability are domains where the phenomena observed are changing overtime. Two main types of technique are used to solve these problems, time integration andspectral methods.

4

The time integration technique advances the solution in time via time steps. Using differentmethods, these techniques discretise the time derivative on the left hand side of equation 1.2to simulate any type of unsteady phenomena. Spectral methods, on the other hand, expressthe quantities in equation 1.3 in terms of harmonic functions. Because of the nature of theharmonic functions they use, spectral methods are however limited to periodic flows.

5

1.2 Elements of the Problematics

1.2.1 Computational Costs of Navier-Stokes Simulations

Currently, the main element limiting the use of CFD in the industry is the computational costof the simulations. Lower simulation time would enable more simulations to be computed andthus reaching better designs through higher number of optimisation iterations and variables.

Simply increasing the available computing power would achieve this objective. However,supercomputer installation is a process with its own challenges. Apart from the cost of thehardware, extensive infrastructures are needed around the supercomputer, such as cooling,security, energy, etc. While the decreasing costs of computational resources means thatthe available computing power for the industry is bound to increase over time, work mustalso be made to find techniques that will solve the Navier-Stokes equations with reducedcomputational resources.

Efficient spatial operators that would reduce the number of iterations needed to obtain con-vergence of a solution is an avenue to reduce the computationnal costs of a simulation,provided that the upgraded spatial operator does not lose in iteration’s cost what it saves initeration number. Time operators are also critical. Since unsteady simulations contain onemore dimension, time, than steady simulations, their computational costs are significantlyhigher. Exceptional costs savings can thus be obtained by using more efficient time operators.

1.2.2 Accurate Solutions of numerical Navier-Stokes Simulation

Accuracy of CFD simulations is also an important issue. Discretisation, artificial dissipationand turbulence modelling all introduce different error sources in numerical flow simulations.

Wind tunnel testing and flight testing are additional experimental tools available to aero-dynamisists. These experimental tools also have their own limitations: wind-tunnel walls,intrusive measures, Reynolds number limit, aero-elastic deformations, etc. They also carrysignificant costs. Any accuracy improvement in computational technique would thus lead tosignificant costs saving on aircraft design programs.

1.2.3 Complex Geometry Treatment

As mentioned previously, space discretisation is a source of error in computational simu-lations. This becomes especially true when simulating high lift configurations containingmultiple bodies (Rumsey et al., 2011). Structured meshes particularly struggle with complexgeometries due to their topology constraints.

6

1.3 Objectives

The current work focuses on developing a framework that will efficiently tackle steady andunsteady flows over geometrically complex airfoils shapes. The framework is NSCODE, aflow solver developed by professor Laurendeau’s research group as a platform to develop novelnumerical methods. Making use of state of the art acceleration techniques, the capabilitiesof NSCODE will be expanded in three main avenues:

• Implement techniques to handle complex geometries;

• Implement efficient steady solver schemes;

• Implement efficient unsteady solver schemes.

1.4 Plan of Thesis

The thesis is split into three main sections: literature review, software development andnumerical results.

As the first section, the literature review covers the main techniques used to obtain efficientspatial and time operators. Different governing equations will be derived and explained aswell as their associated techniques.

The second section, containing the bulk of the work, details the developments of NSCODEconducted within the scope of this work. The framework foundations of NSCODE will bediscussed as they are key to an efficient and lasting software, followed by the developmentsin topological capabilities made to treat complex geometries. Next, the improvements to thesteady solver will cover improvements to dissipation scheme accuracy and spatial operator.The section ends with unsteady flow solver improvements in both time integration and spec-tral methods. Validation and verification phases for each development are also included inthe section.

The last section will contain some challenging CFD simulations made with NSCODE usingthe developments described in the previous section. First, a high-lift optimisation study ismade on the McDonnel Douglas Research Airfoil (MDA) in order to find the optimal flapposition for maximum lift coefficient. Then, a pitching airfoil case under turbulent flow willbe simulated using the different time operators previously developed in NSCODE.

7

CHAPTER 2 LITERATURE REVIEW

This literature review aims to set the theoretical fundamentals of fluid dynamics conceptscovered in this thesis. The first part will focus on developments concerning steady finitevolume solvers for the Euler and Navier-Stokes equations. The second part will focus onunsteady flow solvers and is subdivided into two sections. Time-domain solvers advancingthe solution in time will first be discussed, followed by frequency domain solvers used forperiodic flows.

2.1 Steady Flow Solver

The Navier-Stokes equations, defining the behaviour of viscous flow fields may be expressedin their integral form as

∂

∂t

∫Ω~WdΩ +

∮∂Ω

( ~FC − ~FV )dS = 0 (2.1)

over the domain Ω with a boundary ∂Ω. The conservative variables vector is, in two dimen-sions,

~W =

ρ

ρu

ρv

ρE

(2.2)

the convective fluxes are

~FC =

ρV

ρuV + nxp

ρvV + nyp

ρHV

(2.3)

and the viscous fluxes are

~FV =

0

nxτxx + nyτxy

nxτyx + nyτyy

nxΘx + nyΘy

(2.4)

8

where

Θx = uτxx + vτxy + k∂T

∂x

Θy = uτyx + vτyy + k∂T

∂y(2.5)

(2.6)

nx and ny are the components of outward unit normal vectors to domain boundaries, V isthe contravariant velocity and H the total energy. The pressure p is defined with the perfectgas law

p = (γ − 1) ρ(E − 1

2(u2i

))(2.7)

with γ the ratio of specific heats, and the stagnation enthalpy by

H = E + p

ρ(2.8)

Defining the residual asR (W ) =

∮∂Ω

( ~FC − ~FV )dS (2.9)

equation 2.1 can be approximated as

ΩdWdt

+R (W ) = 0 (2.10)

An implicit system can be obtained from 2.10 by using a backward Euler difference andlinearising around the time level n

(Ω I

∆t + ∂R

∂W

)δW n = −R (W n) (2.11)

Where R is an approximation of the residual R.∂R∂W

can be assembled from the first order approximations of convective and viscous fluxescontribution of a cell and its adjacent neighbours. The upwind flux of Roe(Roe, 1981) areused as a first-order equivalent of the artificial dissipation terms with A and |A| as theconvective flux Jacobian and absolute flux Jacobian respectively

¯F ci+1/2 = 1

2(Fi+1 + Fi)−12 |A|i+1/2(Wi+1 −Wi) (2.12)

The unidimensional notation is used for simplicity. The viscous fluxes are similarly assembled

9

using the thin shear layer (TSL) approximation of the viscous flux Jacobian Av which statesthat the variations in the flow following computing directions parallel a wall boundary arenegligible compared to the variations normal to the wall.

¯F vi+1/2 = −Avi+1/2Wi+1 + Avi+1/2Wi (2.13)

Equation 2.11 diagonal, upper and lower block Jacobians are thus

∂Ri

∂Wi

= [D] = +12 |A|i+1/2 + 1

2 |A|i−1/2 + Avi+1/2 + Avi−1/2 (2.14a)

∂Ri

∂Wi+1= [U ] = −1

2Ai+1 −12 |A|i+1/2 − Avi+1/2 (2.14b)

∂Ri

∂Wi−1= [L] = −1

2Ai−1 −12 |A|i−1/2 − Avi−1/2 (2.14c)

The one dimensional formulation in the i direction is used for simplicity, R is evaluated bythe summation over all computational directions.

2.1.1 Scalar Dissipation

In order to prevent odd-even decoupling and oscillation in regions of strong pressure gradient,it is necessary to add artificial dissipation terms to the finite volume scheme as shown byJameson et al. (1981). From 2.12, the absolute flux Jacobian can be expressed as |A| =T |Λ|T−1. The scalar dissipation scheme (JST) replaces each eigenvalues by the spectral radiusmultiplied by Martinelli’s (Martinelli, 1987) scaling. The absolute flux Jacobian matrix inequation 2.12 is thus effectively replaced by a scalar value

¯F ci+1/2 = 1

2(Fi+1 + Fi)−12λ

(Ai+1/2

)(Wi+1 −Wi) (2.15)

where λ(Ai+1/2

)is the spectral radius of the convective flux Jacobian at the cell face.

Instead of the second difference described in 2.15, Jameson et al. (1981) described an artificialdissipation scheme using a blend of second and fourth difference

ε(2)i+ 1

2(Wi+1 −Wi)− ε(4)

i+ 12

(Wi+2 − 3Wi+1 + 3Wi −Wi−1) (2.16)

10

Where ε(2) and ε(4) coefficients are adapted to the flow using a pressure switch

νi = |pi+1 − 2pi + pi−1||pi+1|+ 2|pi|+ |pi−1|

(2.17)

The switch is used to detect strong pressure gradients, such as in shockwaves, to increase thecontribution of the second difference scheme. The ε coefficients are then found using

ε(2)i+ 1

2= κ(2)max (νi+1, νi) (2.18a)

ε(4)i+ 1

2= max

(0,(κ(4) − ε(2)

i+ 12

))(2.18b)

κ(2) and κ(4) are the artificial dissipation coefficients and typically take the values of 14 and

1128 for the scalar dissipation scheme.

2.1.2 Matrix Dissipation

While the scalar dissipation scheme provides a simple and fast means to apply artificialdissipation, keeping the absolute flux Jacobian’s matrix formulation provides a more accuratescheme. Matricial dissipation scheme (MATD) scales each equation with the appropriateeigenvalues rather than the spectral radius (Swanson and Turkel, 1998),

Using the convective flux Jacobian however leads to numerical difficulties. Near stagnationpoints and sonic lines, some eigenvalues approach zero. In practice, eigenvalues are thuslimited to a fraction of the spectral radius as follows

|λ1| = φ max(|V +

√a2

1 + a22c|, Vnρ(A)

),

|λ2| = φ max(|V −

√a2

1 + a22c|, Vnρ(A)

), (2.19)

|λ3| = φ max (|V |, Vlρ(A)) ,

where

a1 = J−1ξx

a2 = J−1ξy

V = a1u+ a1v (2.20)

and φ is the Martinelli’s scaling (Martinelli, 1987) to account for grid aspect ratio, which is

11

in the I direction

φi = 1 +(ρ (Aj)ρ (Ai)

)β(2.21)

where β is typically set to 0.5.

The Vn and Vl are determined numerically. Typically, values of 0.2 are used and are increasedto 0.4 on coarse grids when using multigrid technique for more robustness.

This yields the eigenvalue vector

Λ = Diag[λ1 λ2 λ3 λ3]

(2.22)

Which is used to recover the absolute convective flux Jacobian matrix via the followingequations (Swanson and Turkel, 1997)

|A| = |λ3|I +( |λ1|+ |λ2|

2 − |λ3|)(

γ − 1c2 E1 + 1

a21 + a2

2E2

)

+ |λ1|+ |λ2|2

1√a2

1 + a22c

(E3 + (γ − 1)E4) (2.23)

E1 =

φ −u −v 1uφ −u2 −uv u

vφ −uv −v2 v

Hφ −uH −vH H

E2 =

0 0 0 0−a1q −a2

1 a1a2 0−a2q a1a2 −a2

2 0−q2 qa1 qa2 0

E3 =

q a1 a2 0−uq ua1 ua2 0−vq va1 va2 0−Hq Ha1 Ha2 0

E4 =

0 0 0 0a1φ −a1u −a1v a1

a2φ −a2u −a2v a2

qφ −qu −qv q

φ =

(u2 + v2

)/2

12

2.1.3 Runge-Kutta Time Integration

A widely used technique to integrate equation 2.10 in time is to utilise an explicit multi stageRunge-Kutta scheme. Jameson (Jameson, 1991) developed an efficient 5 stage scheme takingthe following form

W n,0 = W n−1,K

W n,k+1 = W n,0 − αk∆tΩ R

(W n,k

), where k = 0, K − 1

W n+1,0 = W n,K (2.24)

Where the residual is modified as follows (Jameson, 1983)

R(W n,k

)= Rc

(W n,k

)+Rv

(W n,k

)+ βkR

d(W n,k

)+ (1− βk)Rd

(W n,k−1

)(2.25)

Rc, Rv and Rd being respectively the convective, viscous and dissipative parts of the residual.The α and β stage coefficients used are based on Martinelli’s work (Martinelli, 1987)

αk = (1/4 1/6 3/8 1/2 1 ) (2.26)

βk = ( 1 0 0.56 0 0.44) (2.27)

To ensure stability, timesteps are based on Courant-Friedrichs-Lewy (CFL) number.

Allmaras (Allmaras, 1993) and Pierce et al(Pierce and Giles, 1997), improved the explicitRunge-Kutta integration via the addition of a point-Jacobi preconditionner. The Runge-Kutta stages can be rewritten as

W k+1 = W 0 − αkω[D]−1R (Wk) , where k = 0, K − 1 (2.28)

where ω is the allowable CFL number.

2.1.4 Implicit Residual smoothing

The Implicit Residual Smoothing (IRS) (Jameson and Baker, 1983) method aims at improv-ing the maximum allowable time-step or CFL number by using an implicit operator on theresiduals of explicit schemes. The method requires to solve a tridiagonal matrix for eachresidual of the conservative variable on each computational directions using the following

13

formulation− εI ~R∗

I−1 + (1 + 2εI) ~R∗I − εI ~R∗

I+1 = εI ~RI−1 (2.29)

with R∗ the smoothed residual. ε is defined by Swanson and Turkel (1997) as

εI = max

14

( NN∗

φ

1 + rI

)2

− 1 , 0

(2.30)

where φ is the same coeficient used for Martinelli’s scaling (see equation 2.21), rI is the ratioof spectral radii

rI = λJλI

(2.31)

and NN∗ is given the value 2.0.

Using a tridiagonal solver on equation 2.29, the method is computationally inexpensive andincreases the maximum allowable CFL number.

2.1.5 Block lower-upper symmetric Gauss-Seidel scheme

Yoon and Jameson(Yoon and Jameson, 1986; Jameson and Yoon, 1987; Yoon and Jameson,1988) developed a lower-upper symmetric Gauss-Seidel (LU-SGS) scheme to solve equation2.11. The scheme relies on a forward and backward sweep through the domain

[D]δW 1i = −R (W n)i − [L]δW 1

i−1 (2.32a)

[D]δW 2i = −R (W n)i − [L]δW 1

i−1 − [U ]δW 2i+1 (2.32b)

Equation 2.32 uses the full matrix operators defined in 2.14 to enhance convergence rateas described in Wright et al. (1996). In the first sweep, the δW information previouslycomputed within the sweep on other cells are used with the [L] block. Second sweep uses δWof previously computed cells of the second sweep with the [U ] block and the informationsof the previous sweep for the cells not yet computed on the current sweep for the [L] block.The state vector is updated once the sweeps are completed

W k+1 = W k + ωδW (2.33)

Where ω is the LU-SGS relaxation factor typically ranging between 0.5 and 1.0 (Cagnoneet al., 2011).

14

It is possible to increase the number of sweeps in order to reduce the block tridiagonal matrixinversion errors as shown in Cagnone et al. (2011). This is done by alternating betweenforward and backward sweeps using the latest available solution for δW . For example, thethird and fourth sweeps operations would be

[D]δW 3i = −R (W n)i − [U ]δW 2

i+1 − [L]δW 3i−1 (2.34a)

[D]δW 4i = −R (W n)i − [L]δW 3

i−1 − [U ]δW 4i+1 (2.34b)

2.1.6 Multigrid

Probably the most effective convergence acceleration method for subsonic and transonicaerodynamic CFD simulations is the multigrid method. Applied to CFD by South Jr andBrandt (1976) and further developed by Jameson (1986), the method is now used in manyCFD solvers. However, its implementation is difficult and optimal convergence rates are notalways obtained.

The principle of the method relies on the fact that fine meshes eliminate high frequency errorsrapidly but are not effective to remove low frequency errors. To address this issue coarsergrids, obtained by removing every other point from the original grid, are used. Therefore,some low frequency errors of the fine grids become high frequency errors for the larger cellsof the coarse grids and are thus rapidly removed. By transferring the solution from the finegrids to the coarse grids to compute corrections, and then applying back the corrections tothe fine grid, a wider range of error frequencies are removed effectively. The cost of thetechnique is also relatively low since each level of coarse grid contains 4 times less cells in 2Dproblems and 8 time less in 3D problems, their computation cost scales by the same factors.

2.2 Unsteady Reynolds-Averaged Navier-Stokes solvers

Unsteady flows are present in many situations, such as turbo-machinery flows, helicopter inflight and aircraft stability studies. As such, time accurate schemes are required to correctlysimulate such flows. Solver accelerators described previously, such as multigrid and local timestepping schemes, sacrifice time accuracy to achieve faster convergence. If those acceleratorsare to be kept, new schemes must be implemented to compute time accurate simulations.

15

2.2.1 Dual Time-Stepping

Jameson (Jameson, 1991) developed the dual-time stepping scheme in order to benefit fromsteady state solver acceleration while maintaining time accuracy.

Using a second order backward difference to evaluate the time derivative in equation 2.10

32∆t [W

n+1Ωn+1]− 2∆t [W

nΩn] + 12∆t [W

n−1Ωn−1] +R(W n+1

)= 0 (2.35)

where n denote the real time step number.

Equation 2.35 can be treated as a modified steady state problem using timesteps in fictioustime t∗

dW

dt∗+R∗ (W ) = 0 (2.36)

where the modified residual R∗ (W ) is defined as

R∗ (W ) = 32∆tW + 1

Ωn+1

(R (W )− S

(W n,W n−1

))(2.37)

with the source term

S(W n,W n−1

)= 2

∆t (W nΩn)− 12∆t

(W n−1Ωn−1

)(2.38)

Extending the steady LU-SGS space solver scheme with dual time stepping is straightforward.Using a backward difference to conserve second order time accuracy instead of an Eulerdifference in 2.11 (Hirsch, 2007), one obtains the following

(3Ω2

I

∆t + ∂R

∂W

)δW n = −R (W n) (2.39)

From there, keeping ∆t as the specified step for time integration, the diagonal matrix con-tribution in equation 2.14 is now

∂Ri

∂Wi

= [D] = +12 |A|i+1/2 + 1

2 |A|i−1/2 + Avi+1/2 + Avi−1/2 + 3Ω2

I

∆t (2.40)

The LU-SGS other operations remain the same as their steady solver counterparts.

16

2.2.2 Non-Linear Frequency Domain

Many aerodynamic computations aim at simulating unsteady problems which exhibits re-peating patterns in time. Such periodic problems are interesting in their periodic steadystate, where any variation in the periodic behaviour of the flow is damped out. Since tradi-tional unsteady calculations such as the Dual Time-Stepping method are time accurate, thatis the solver integrates the solution in time from an initial state, the solver must computethe transient state of the flow from initial condition to periodic steady state.

The Non-Linear Frequency Domain Method (NLFD) was proposed by Hall et al. (2002) andmodified by McMullen et al. (2001, 2002) to directly solve flow solutions to periodic steadystate. Relying on the assumption of periodic flows, Fourier series can be used to representthe solution. Solving the solution as a steady periodic flow in the frequency domain alsoallows for the use of steady solver’s acceleration method, including multigrid and implicitresidual smoothing (IRS). The computational cost of the NLFD method thus scales linearlywith the number of modes computed with roughly the cost of (2 · Nmodes + 1) times steadysimulation.

Governing equations

From equation 2.10, the state variable vectors and residual vectors are formulated as Fourierseries containing N modes

W =N2 −1∑

k=− N2

Wkeikt (2.41a)

R(W ) =N2 −1∑

k=− N2

Rkeikt (2.41b)

(2.41c)

To yield

Ω d

dt

N2 −1∑

k=− N2

Wkeikt

+N2 −1∑

k=− N2

Rkeikt = 0 (2.42)

Where Wk and Rk are the Fourier coefficients of the state vector and residual vector respec-tively.

17

A solution of equation 2.42 is

ikΩWk + Rk = 0, k = −N2 ,N

2 − 1 (2.43)

We can thus describe a modified residual in the frequency domain as

R∗k = ikΩWk + Rk = Ik (2.44)

We then obtain a new equation that we can march in pseudo time τ in order to converge toa solution

ΩdWk

dτ+ R∗

k = 0 (2.45)

McMullen (McMullen et al., 2001, 2002) showed that convergence acceleration techniquessuccesfull for equation 2.10 can also be used on 2.45. Using the modified NLFD residual R∗

instead of the steady residualR, multi stage Runge-Kutta scheme, implicit residual smoothingand multigrid techniques can be applied to equation 2.45.

2.3 Framework Development

Framework planning is a growing issue in the CFD software development community. Ageingprograms becoming harder to maintain because of tool obsolescence, poor programming prac-tices and lack of flexibility. These factors ultimately lead to software performance decrease,longer familiarisation time for new developers trained in recent computational techniques,and increased programming errors (bugs).

For these reasons, many institutions are undergoing the development of brand new frame-works in order to maintain efficiency of currently developed software. For example, ONERAlaunched the elsA project (Thibert) to provide a common platform for all CFD developmentsof the ONERA’s projects. Another example is the American Department of Defence(DoD)’sCREATE program (Arevalo et al., 2008) which aims at regrouping the DoD’s design toolsfor ships and aircrafts while making optimal use of high performance computing resources.

The next section discusses in details the approach taken for the present software develop-ments.

18

CHAPTER 3 SOFTWARE DEVELOPMENTS

3.1 Framework

An important yet often undervalued step when planning any software development project isto choose an appropriate framework. Extensive projects typically involving many program-mers and developments spanning over years must be carefully designed. Many frameworkrelated problems can undermine the long term usage and updating of a software. For thesereasons, extra care was given to the choice of programming practices and used tools in thedeveloped software.

The CFD platform used for the current work is NSCODE, a structured mesh based cell-centreRANS and URANS flow solver (Pigeon et al., 2014). The software is currently developed atPolytechnique Montreal.

Two programming languages are used to develop NSCODE, C and Python. The C languageis currently one of the fastest languages apart from machine language for software executionwhile Python is known for its flexibility and easy readability. Computationally expensiveparts of the software are thus developed in C to obtain optimal performance while high leveltasks requiring low computational resources are written using Python. The interface betweenthe two languages is handled via the f2py library, currently included in the SciPy packagewhich includes numerous Python libraries to enhance the scientific computing capabilities ofthe language. Typically, Python scripts handle inputs and outputs as well as calling the mainmodules of the flow solver while the core CFD functions are compiled from C source code.

Figure 3.1 shows an example of the execution flow of an NSCODE simulation. Each noderepresents one of the main steps called by the Python script.

Since NSCODE is used and developed by a growing number of students, the usage of arevision control software is necessary to keep track of the code modifications from multipledevelopers. Such software allows for easy merging between two versions as it handles filedifferences and highlights the files affected by multiple changes. Revision control softwarealso ease any debugging process by keeping track of every change made to the source code.Amongst the available revision control software, Mercurial was used for NSCODE due to itssimplicity of use. Using a centralised repository on a server, each developer can upload anddownload the latest changes in order to keep the whole development team up to date.

19

Input parameters specifications

Input file creation

NSCODE initialisation

Topology creation

Flow Solver

Output file management

Figure 3.1 Flow chart of steady NSCODE execution

3.2 Topology Capabilities Improvements

Flow simulations around complex geometries is one of the challenges of a structured flowsolver. Indeed, meshes used by such solvers are Cartesian in the topological space. Whilesuch meshes are appropriate to mesh single element airfoils using "O" or "C" type topologies,they are illsuited for geometries containing more than one element such as wing-flap, aileronor spoiler configurations.

To simulate flows around such geometries, a multi block approach is required. The idea is todecompose the domain into different blocks, each block being covered with its own structuredmesh. The structured solver can then solve the flow on each blocks. Although each blockis solved independently, communication between adjacent blocks must still be assured inorder to obtain a valid solution over the whole domain. To allow blocks to communicatethe information at their boundaries to adjacent blocks, the implementation of a connectionboundary condition is required.

The topology flexibility of the solver is then extended by the implementation of the Chimeramethod. This method allows to treat overset meshes, removing the meshing constraint tohave adjacent blocks with matching points at their boundaries (sometimes refered to as one-to-one connections). Overset meshes overlap each other arbitrarily, such that communicationthrough each blocks is assured by interpolations rather than boundary conditions at blockedges. This greatly simplifies mesh generation since each geometrical element of a problemcan be meshed separately of the others, resulting in higher quality meshes.

20

3.2.1 Multiblock Solver

The multi-block topology extension contains three main tasks: developing a topology inputformat to describe blocks topologies, treating the connections between blocks and adaptingthe solver to treat more than one block.

To enable flow solvers to handle a multiblock mesh, one must develop a way to describe thearrangements of the block relative to one another. In single block meshes, since most of thetopologies being of the "O" or "C" type, the topology treatment can be simply hard-codedfor these two possibilities. However, for multiblock meshes, we must adopt an efficient inputtopology file format to describe how the various blocks are connected to each other.

An example of a sample topology input file is showed in Annex A. After a title, the topologyinput contains the number of blocks and the maximum dimensions of each block in eachcomputational directions. Then, a pattern is repeated to describe each block. The first lineof the pattern states the block number and the number of different boundaries it contains.Note that a block can contain more than four boundaries if a boundary does not cover anentire block edge. Then, each boundary condition of the block is described using an additionalline. First, the type of boundary is specified using a three character code. Then, the positionof the boundary is specified using the starting and ending indices of each computationaldirections. The last field on each line is used with wall boundary condition to specify whichbody is associated with the boundary condition. The connection boundary condition takesan additional line to specify which block the boundary is connected to and the associatedlocation of the boundary on the other block.

Once the boundaries can be accurately described with the input file, the software must beupdated to handle the connections between the blocks. The algorithm uses phantom cellsto enforce boundary conditions. Two layers of phantom cells are used around each block topreserve the stencil which uses third order dissipative scheme throughout the computationaldomain. The block connection boundary conditions are handled by copying the connectingblocks information into the phantom cells. Using the topology input information, two layersinside the connecting block’s domain along the boundary are found and associated withthe proper phantom cells. This methodology allows for continuous transfer of informationthroughout the block’s connecting faces, leading to the same solution as an equivalent singleblock resolution through machine accuracy.

The topology treatment also allows for completely generalised connections between blocks.Blocks can thus connect along any of their faces, including connections along different com-putational dimensions, that is an I normal face connecting with a J face, and of opposite

21

directions, that is when the cells index increase along the boundary in the first block whilethey decrease along the boundary of second one.

Other than implementing the additional boundary treatment, solver adaptation to multiblock meshes is straight forward. Each block can be treated as a separate computational do-main and as such each subroutines of the software only need to be nested inside a loop overeach block. This method provides an ideal way of using coarse grain parallel computing, sep-arating the computation of each blocks along different CPU’s. In the current framework, theshared memory tool OpenMP is used for its implementation simplicity although performanceis limited to the number of cores sharing the memory.

3.2.2 Overset mesh compatibility

Overset mesh capability, also called chimera method, is the capacity of the solver to handlemultiblock meshes were block boundaries do not lie adjacent to one another. Instead, oversetgrids arbitrarily overlap each other. While this technique greatly reduces meshing constraintson complex geometries, the chimera boundaries between blocks must be handled by the solver.

Typically, a preprocessor is used to treat chimera grids in order to determine which cellswill be used to calculate the solution (donor cells), which ones will receive information fromother blocks to ensure connectivity (fringe cells) and which ones should be discarded (Holecells). Also, for each fringe cells, the preprocessor finds the donor cells used to interpolatethe solution and their interpolation weight. Some CFD software developers opted to de-velop standalone chimera preprocessors, such as NASA’s Pegasus (Suhs et al., 2002). Thebenefit of standalone preprocessors is the flexibility of working with any flow solver usinga standard communication format, such as the CGNS format. The drawback however, isthe communication overhead between the chimera preprocessor and the flow solver. Whilethis overhead may be acceptable for steady simulations, it can become an issue when thepreprocessing must be done repeatedly such as for unsteady simulations with moving grids.For this reason, the preprocessor, developed within the research laboratory of professor EricLaurendeau at Polytechnique Montreal (Pigeon, 2015; Levesque et al.), is included in theNSCODE framework. The Chimera preprocessor is added in the workflow after the topologycreation step in figure 3.1. A very detailed account of the preprocessing steps can be foundin (Pigeon, 2015)

Once the chimera preprocessing is done, a minimal amount of work is required to adapt theflow solver. The first step is to add the chimera interpolation when updating the boundaries.Since all the information required to interpolate the solution on fringe cells is already known,this step consists of looping through the fringe cells and assigning it a state vector consisting

22

of a weighted sum of its donor cells’ state vectors. The second step is to prevent the solutionupdate of fringe and hole cells when updating the solution. This avoids the overwriting ofthe interpolated values of fringe cells.

Multigrid Issues with Overset Meshes

A major limitation of the chimera method is that convergence to machine accuracy is notachieved using geometric multigrid. The geometric multigrid technique relies on removingone point out of two in all computationnal directions to produce coarser grids. While thistechnique works well on one-to-one meshes where points on the boundaries are not affected,the algorithm fails when used on chimera grids. Indeed, chimera boundaries can be positionedarbitrarily in the domain. Some point forming the boundary are then removed during themultiblock process. This results in a change in the boundary conditions location from one gridto the other, changing problem definition. Multigrid corrections passed from a level to theother will thus be evaluated for slightly different problems preventing solution convergenceto machine accuracy.

This issue can be illustrated using a case of Euler flow around a NACA0012 airfoil. The Machnumber is 0.7. The mesh used is the same as the Euler mesh used in section 3.3 cropped atthe 65 closest rows to the wall. A cartesian background mesh extending 25 chord lengths ineach directions is overlaped to provide far-field resolution.

Figure 3.3 shows the convergence obtained when using two level of multigrid versus no multi-grid levels. The simulation using multigrid starts converging slightly faster than the otherone which doesn’t uses multigrid but floors out after reducing the density residual by aboutthree orders of magnitude.

23

Figure 3.2 NACA0012 overset mesh for testing of multigrid technique on chimera grids

Figure 3.3 Convergence of the NACA0012 airfoil test case with overset mesh

24

Transonic Biconvex Airfoil

The transonic biconvex airfoil problem is a critical case to test the chimera method sincea shockwave is located across the boundary of the two meshes (Soni et al., 2012). Anyinterpolation errors can thus be amplified. The geometry consist of two NACA0012 airfoilsstaggered horizontally and vertically by 0.5 chords length. This test case is simulated in aninviscid flow with a free stream Mach number of 0.7 and zero degree angle of attack. Foreach airfoil, a 128x128 Euler grid generated by Vassberg and Jameson (1981) was used for atotal of 32768 cells. No background meshes were used. A buffer of three layer was used withbilinear interpolations.

As can be seen in figure 3.4, the solution yields smooth pressure contours throughout thedomain with seamless passage through the chimera boundaries. Figure 3.5 shows the pres-sure coefficient (CP ) distribution of the upper and lower airfoils compared to Morinishi’s(Morinishi, 1992) results with non-body-fitted Cartesian grids.

Figure 3.4 Staggered NACA0012 pressure isobars with grid M = 0.7

25

(a) lower airfoil (b) upper airfoil

Figure 3.5 Staggered NACA0012 pressure coefficients M = 0.7

McDonnell Douglas 30P30N Airfoil

The McDonnell Douglas 30P30N Airfoil (MDA) is another interesting test case since it show-cases the ability of chimera grids to handle complex geometries with ease. It is also a populartest case in the literature (Liao et al., 2007; Wang and Parthasarathy, 2000; Mavriplis andMani).

The chimera grids for each element are obtained with a grid generator (NSGRID), developedwithin the research laboratory of professor Laurendeau at Polytechnique Montreal, whichuses a blend of parabolic and elliptic approaches (Hasanzadeh et al., 2013). The flap and slatmeshes each contain 33000 grid cells whereas the main element mesh contains 132000 gridcells. No background mesh is necessary for this case since the flap and slat meshes are bothlocated inside the main mesh. The first cell is located at 10−6 chord off the airfoil surfaceand an expansion ratio of 1.15 is used to strech the gridlines normal to the wall. Clusteringis used at leading and trailing edge regions.

The basic 30P30N MDA is used to compare the solution obtained with chimera meshes tothe ones obtained with a one-to-one multi-block mesh containing 256 000 grid cells withsimilar characteristics. Figure 3.6 shows both meshes. Experimental data is also plotted tocompare the CP distributions. The free stream flow conditions are Mach 0.2, Re 9× 106 and16.21 angle of attack. The Spalart-Allmaras turbulence model is used.

Figure 3.7 shows the convergence obtained while solving the flow. The one-to-one resolutionswere made using full approximate storage with 300 iterations on the 3rd grid level, 6000 onthe 2nd grid level before solving on the finest grid. Although the 2nd one-to-one grid levelshows better convergence rate than the chimera grid due to the usage of multigrid acceleration

26

technique, the finest level is unable to converge.

Figure 3.7 also shows that the pressure distribution obtained with the chimera implementa-tion of NSCODE matches both the one-to-one resolution of NSCODE and the experimentaldata on the slat and main element. However, the chimera resolution seems to produce anearlier separation at the flap. This behaviour can be explained by our initialization fromfree stream conditions which tend to produce early stalls on high-lift devices (Rumsey et al.,2011; Deloze and Laurendeau).

Figure 3.6 Chimera and one-to-one grids used with NSCODE to compute the McDonnellDouglas 30P30N Airfoil test case

Figure 3.7 Convergence and pressure distribution obtained on the 30P30N airfoil

27

3.3 Steady Solver Improvements

3.3.1 Matrix dissipation

The matrix artificial dissipation scheme (MATD) implementation is completed using theabsolute flux Jacobian described in 2.23. To ensure that the scheme is effective and accurate,the values of Vn and Vl used in equation 2.20 were kept at 0.2 on fine grids and 0.4 on coarsegrids. Since The MATD scheme is less dissipative than the scalar dissipation scheme bydefinition, the artificial dissipation coefficients κ(2) and κ(4) are increased from 1

4 and 1128 to

12 and 1

32 respectively.

3.3.2 Point Jacobi

The block implicit Point Jacobi preconditioner was implemented as described in equation2.28. The method relies on transforming the implicit system of equation 2.11 into a blockdiagonal, ignoring the off diagonal contributions of the [U ] and [L] blocks in equation 2.14.This simplification limits the implicit behaviour within a cell, involving a single four by fourmatrix inversion for each cell in 2D and five by five inversion in 3D. Since the contributionsof other cells to the system is not used, the explicit Runge-Kutta multistep scheme can beused by adding the matricial operator to update the solution as described in equation 2.28.The LU decomposition technique is used to inverse the [D] matrix.

In order to have a good approximation of the value of ∂R∂W

from equation 2.11, Jacobiansmust be evaluated for the dissipative and viscous fluxes. The absolute flux Jacobian arereused from the matrix-dissipation scheme described above. However, several formulationsfor the viscous flux Jacobian are used throughout the CFD community. The thin shear layer(TSL) approximation is widely used to simplify the Jacobian formulation. While the TSLapproximation is meant to be used normal to the wall boundary, it can be applied in allcomputational directions (Blazek, 2005). For further simplification, one can also replace theviscous Jacobian with a diagonal matrix containing the viscous spectral radii (Sharov andNakahashi, 1997). A third formulation of the viscous Jacobian is the one used in NSMB, a3D structured flow solver developed through European partnerships, as described in Weber(1998). After testing each formulation, it was found that the NSMB Jacobian was the oneyielding the best performance and robustness. Each formulation is detailed in Annex B.

28

3.3.3 LU-SGS scheme

The LU-SGS implicit solving scheme was added to the software in order to achieve higherperformance on stretched grids. Many challenges arose and lead to minor changes in theinitial scheme in order to achieve desired performance.

As for the Point Jacobi scheme, the absolute flux Jacobian and viscous flux Jacobian mustbe evaluated at each cell faces. Their formulation in the LU-SGS solving scheme is the sameas in the Point Jacobi scheme. In addition, the convective flux Jacobian must be evaluatedusing the flow properties at each cell centre for the [U ] and [L] blocks in equation 2.14, itsformulation is based on Pulliam and Steger (1985).

Although it is possible to let ∆t→∞ in equation 2.11, additional robustness can be addedto the scheme for challenging computations by taking a finite ∆t. This fictitious time step’svalue is determined in the same way as the explicit time step in a purely explicit time steppingscheme. The new term is added to the diagonal of the [D] matrix of equation 2.14. Theadded diagonal dominance helps stretched meshes to converge properly to a solution at theexpense of convergence rate. Setting the CFL number to a very high value will set this termto zero.

In order to compute the LU-SGS domain sweeps, the order of the cells on which equation 2.32is evaluated is important. On the upward sweep, the cell evaluation order must ensure thatthe lower cells information is already computed and available while on the downward sweep,the upper cells must be computed first. This problem is treated by sweeping the domainwith successive diagonal paths. Figure 3.8 illustrates the diagonal domain sweeping of anupward LU-SGS sweep. The solid dots represent the cells where δW correction is alreadycomputed by the current sweep, the empty dot is the cell where δW is currently computedand the other cells are the ones where the results of the previous sweep will be used. Thedouble arrows represent the order in which the domain is covered. In other words, on figure3.8, the empty dot is the cell where the matrix [D] is inverted, the informations of a soliddot cell will multiply the lower matrix [L] and an empty cell will multiply the upper matrix[U ] to find the correction δW of the current sweep on the empty dot cell.

Another important aspect to consider when using the LU-SGS schemes on multiblock meshesis the availability of the information in phantom cells when at the edges of block connec-tions. Without boundary treatment, the off diagonal blocks of the cells alongside the blockconnections would be lost, reducing the effectiveness of the implicit scheme. To obtain acomplete domain implicit treatment, the δW n obtained after each upward and downwardsweeps described by equations 2.32 and 2.34 must be passed through the connecting bound-

29

j

i

Figure 3.8 Node treatment order of the LU-SGS upward sweeps

ary conditions as described in section 3.2.1.

3.3.4 Validation and Verification

Assessment of the effectiveness of the numerical method was made on numerous test cases.First we validate the matrix artificial dissipation scheme and the space order accuracy of thecode. Then, we verify that the new Point Jacobi and LU-SGS solvers converge to machineaccuracy without affecting results. In addition, the efficiency of the parallell implementationwill be calculated. Also, the LU-SGS scheme will be used with two and four sweeps to comparethe performance trade-offs of adding more sweeps in the LU-SGS schemes as described inequation 2.34.

In each test cases, implicit residual smoothing is used for the Explicit Runge-Kutta and PointJacobi Runge-kutta methods. In addition, four levels of multigrid are used on each meshusing a W cycle with two fine grids resolutions before and after the cycle.

NACA0012

The first test case chosen is a simple NACA0012 airfoil in an inviscid Euler flow. The 128x128NACA0012 Euler O-mesh from Vassberg and Jameson (1981) is used. The free stream Machnumber is 0.8 and the angle of attack is 1.25 . The CFL number and LU-SGS relaxationfactors(ω) used for each numerical method are listed in table 3.1.

Figure 3.1 shows the convergence rates of the various techniques with respect to multigridcycles and time. As expected, the LU-SGS method with four sweeps (LU-SGS4) achievedmachine accuracy convergence with the lowest number of iterations. The LU-SGS4 scheme

30

also proved to be the fastest to obtain machine accuracy in term of CPU time, just short ofthe explicit Runge-Kutta scheme with matrix dissipation. A surprise with this test case isthe poor performance of the Point Jacobi preconditioner. It fared no better than the ExplicitJST scheme in terms of convergence rate per iterations while taking more CPU work periteration.

Figure 3.11 show the Cp distribution over the airfoil. As can be expected, oscillations inthe region of the shockwave appear when using the MATD scheme. This is due to the lessdissipative nature of the scheme compared to the JST scheme which smoothes out theseoscillations. In order to assess the effect of the dissipation sheme on these oscillations, thetest case is computed using different values of Vn and Vl (0.4, 0.6 and 0.8). Since higher valuesof Vn and Vl produce absolute jacobian with higher eigenvalues, as dictated by equation 2.20,and thus stronger dissipation, lower oscillations amplitude are obtained.

Table 3.2 lists the aerodynamic coefficients obtained. As expected, the lift and drag coeffi-cients were influenced by the change in artificial dissipation scheme since it changes the righthand side of equation 2.11. The choice of solving scheme however does not influence theresults at machine accuracy since they only affect the left hand side of equation 2.11.

Table 3.1 Euler NACA0012 allowable CFL number/ω

Solving scheme Dissipation scheme CFL/ωExplicit Runge-Kutta JST 7.5Explicit Runge-Kutta MATD 7.5Point Jacobi Runge-Kutta MATD 7.5LU-SGS 2 sweeps MATD 0.6LU-SGS 4 sweeps MATD 0.6

Table 3.2 Euler NACA0012 lift and drag coefficients

Solving scheme Dissipation scheme CL Cdp CdfExplicit Runge-Kutta JST 0.357869 0.022962 0.000000Explicit Runge-Kutta MATD 0.354330 0.022490 0.000000Point Jacobi Runge-Kutta MATD 0.354330 0.022490 0.000000LU-SGS 2 sweeps MATD 0.354330 0.022490 0.000000LU-SGS 4 sweeps MATD 0.354330 0.022490 0.000000

31

Figure 3.9 Residual convergence of the Euler NACA0012 test case with respect to iterationsand CPUtime

32

Figure 3.10 Mach contours of the Euler NACA0012 test case

Figure 3.11 Pressure distribution of the Euler NACA0012 test case

33

This case was also used to assess the parallel computing efficiency of the OpenMP imple-mentation. The 128x128 grid used previously is split into 8 blocks of equal size (32x64).Simulation using 1, 2, 4 and 8 processors are launched using both runge-kutta explicit andLU-SGS scheme. The time to complete a multigrid cycle, computed over the average ofa hundred cycles, is used to calculate the speedup obtained with the different numbers ofprocessors.

Figure 3.12 shows the speedup curve obtained with NSCODE. It is observed that at eightprocessors, the efficiency of the paralellisation is of 71%. Using Amdahl’s law, we obtain aparallelised portion of NSCODE of 94.6%.

Figure 3.12 Parallel speedup factor of NSCODE

Following a similar procedure as Vassberg and Jameson (1981), the Euler NACA0012 testcase is used to assess the order of accuracy of the steady solver. In addition to the 128x128grid used previously, refined grids provided by Vassberg and Jameson (1981) of size 32x32,64x64, 128x128, 256x256, 512x512, 1024x1024 and 2048x2048 to capture grid convergenceof the solution. From these, the order of accuracy p and continuum estimates C∗

L and C∗D

are extracted via the procedure described in Vassberg and Jameson (1981). The freestreamflow conditions are unchanged from the test case except for the Mach number of the subsonicstudy which is decreased to 0.5.

Table 3.3 shows the orders of convergence, the lift and the drag coefficients of established CFDsolvers and NSCODE. In the subsonic range, NSCODE is below second order convergencefor the lift but shows hyper-convergence for the drag (Pigeon et al., 2014). Although not

34

shown in the table, a similar hyper-convergence trend was observed by Vassberg and Jameson(1981) with FLO82 for the transonic non lifting case. In the transonic regime, both lift anddrag exhibit a first-order-accurate character which is expected due to the artificial dissipationscheme first order switch at shock waves. Continuum estimates for lift and drag coefficientsfall within the range of the other CFD methods.

Table 3.3 Convergence order and continuum estimates for subsonic and transonic NACA0012Euler solutions

M = 0.5 M = 0.8O(CL) O(CD) C∗

L C∗D O(CL) O(CD) C∗

L C∗D

FLO82 2.107 1.805 0.1803 0.000000050 1.151 1.118 0.3562 0.02268Overflow 1.162 0.820 0.1798 0.000010030 0.438 0.785 0.3517 0.02245CFL3Dv6 1.153 2.060 0.1804 -0.000000134 0.289 0.960 0.3516 0.02267NSCODE 1.526 2.654 0.1794 0.000019333 0.928 0.950 0.3529 0.02251

Figure 3.13 Convergence order for subsonic and transonic NACA0012 Euler solutions

RAE2822

The RAE2822 test case is chosen to test the numerical techniques using RANS equations.The free stream Mach number is 0.73 with an angle of attack of 2.79 and a Reynolds numberof 6.5 million. The turbulence model used is Spalart-Allmaras using ten model iterationsper multigrid cycle with six ADI sub-iterations as in Cagnone et al. (2011). The mesh usedcontains 18432 cells and the y+ of the first cells to the wall is inferior to 1. To enhancecomputing speed, the mesh is split between 6 blocks.

35

Table 3.4 RAE2822 allowable CFL number/ω of NSCODE compared to Cagnone et al. (2011)

Solving scheme Dissipation scheme CFL/ω Cagnone et al. (2011)Explicit Runge-Kutta JST 7.5 -Explicit Runge-Kutta MATD 5.0 5.0Point Jacobi Runge-Kutta MATD 5.0 7.0LU-SGS 2 sweeps MATD 0.4 0.9LU-SGS 4 sweeps MATD 0.4 0.9

Figure 3.14 RAE2822 grid

One of the difficulties of the test case is the loss of robustness due to the less dissipativeMATD scheme. Since the mesh used contains stretched cells in the boundary layer and wakeregions, the reduced robustness of the MATD scheme forced to reduce the CFL number from7.5 to 5.0 for simulations using the scheme as summarised in table 3.4. This explains thegood performance of the explicit JST scheme in convergence rate versus time seen in figure3.15. Another observation is that the Point Jacobi preconditioner which seemed ineffectivefor the previous test case now yields the best convergence rate for the number of iterations.It seems that the point implicit preconditioning is beneficial when used on stretched cells.Note that this behavior has also been observed by Cagnone et al. (2011).

Table 3.5 RAE2822 lift and drag coefficients


36

Figure 3.15 Residual convergence of the RAE2822 test case with respect to iterations andCPUtime

37

Figure 3.16 Mach contours of the RAE2822 test case

Figure 3.17 Pressure distribution of the RAE2822 test case

38

NLR7301

The objective of the NLR7301 test case is to asses the performance of each solver over a twoelement airfoil. The free stream Mach number is 0.185 with an angle of attack of 13.1 and aReynolds number of 2.51 million. As with the previous test case, the turbulence model used isSpalart-Allmaras using ten model iterations per multigrid cycle with six ADI sub-iterations.The mesh, provided by Bombardier Aerospace (Cagnone et al., 2011), contains 145k cellsbetween 9 blocks and the y+ of the first cells to the wall is about 0.5. Since the angle ofattack is relatively high, computing stability is improved by gradually increasing (ramping)the simulation angle of attack from 3.0 to the final value of 13.1 using 400 multigrid cycles,as done in Cagnone et al. (2011).

Table 3.6 NLR7301 allowable CFL number/ω of NSCODE compared to Cagnone et al. (2011)

Solving scheme Dissipation scheme CFL/ω Cagnone et al. (2011)Explicit Runge-Kutta JST 7.5 -Explicit Runge-Kutta MATD 5.0 2.0Point Jacobi Runge-Kutta MATD 5.0 2.0LU-SGS 2 sweeps MATD 0.2 0.9LU-SGS 4 sweeps MATD 0.2 0.9

Figure 3.18 NLR7301 grid

The results shown are taken after 2000 iterations of multigrid cycles. The first surprise withthis test case is that while the Explicit Runge-Kutta scheme showed convergence rate, thePoint Jacobi preconditionner and the LU-SGS schemes picked up unsteady modes and hadtheir convergence stalled as illustrated in figure 3.20. This observation is contrary to whatCagnone et al. (2011) observed on the same test case. Since convergence of the residual to

39

Figure 3.19 Mach contours and pressure distribution of the NLR7301 test case at 13.1 angleof attack

machine accuracy is not reached, the values of the aerodynamic coefficients differ from onesolver to the other in table 3.7. When simulating the same test case at 5 angle-of-attackhowever, the flow remains completely attached and all solvers show good convergence ascan be seen in figure 3.21. Table 3.8 also show accordance on the aerodynamic coefficientsbetween the different solvers used.

Figure 3.20 Residual convergence of the NLR7301 test case with respect to iterations andCPUtime at 13.1 angle of attack

40

Table 3.7 NLR7301 lift and drag coefficients at 13.1 angle of attack


Figure 3.21 Residual convergence of the NLR7301 test case with respect to iterations andCPUtime at 5 angle of attack

Table 3.8 NLR7301 lift and drag coefficients at 5 angle of attack


41

3.4 Unsteady solvers

3.4.1 Dual Time-Stepping

The addition of the Dual Time Stepping scheme (DTS) to the framework required minorchanges to the software. In addition to a simple source term addition, the implementationtook advantage of the two loops over pseudo time and real time. The loop over pseudo timeis very similar to the steady flow solver’s pseudo time marching. As such, the steady flowsolver was modified to calculate the additional terms in the residual described in equation2.37.

Since the loop over real time requires several user inputs and little computation costs, it isplaced inside the python script. This choice easily gives users the flexibility to launch complexsimulations. Indeed, it is often observed that initialising an unsteady problem by using a fewmultigrid cycles of steady flow solvers can reduce the simulation time put on the transientevolution of the flow. This methodology also allows to start an unsteady simulation with alarge timestep to quickly observe periodic steady state and then change to a finer timestepto study the flow with good time accuracy. Figure 3.22 is an example of flowchart for anunsteady simulation using the steady solver for initialisation.

3.4.2 Non Linear Frequency Domain Solver

As described in section 2.2.2, the Non Linear Frequency Domain method is a very efficientmethod for simulating periodic flows. However, the implementation of such a solver requiresextensive software architecture adaptation. In order to obtain the Fourier coefficients de-scribed in equation 2.41, several solutions along the time period are needed depending on thenumber of modes in the Fourier series. The flow solver must compute 2N + 1 solutions inparallel at different points in time, or time samples, in order to complete the Fourier trans-forms, N being the number of modes. Time thus effectively becomes an additional dimensionof the computational domain. It is important to note that computational costs in CPU timeand memory requirements also scale linearly with the number of time samples.

The figure 3.23 represent the mains steps of the NLFD solver. From the set of Fouriercoefficient of the conservative variables W , an inverse Fourier transform is used to get theconservative variables W at each time sample. Then, the residual as defined in equation 2.9is calculated from the state vector at each time sample. A Fourier transform is then used toobtain the Fourier coefficient of the residuals R. From the Fourier coefficients of the residualand of the state vector, the modified residual R∗ can be computed. Iterations are then made

42

by updating the Fourier coefficients of the solution W using R∗. Since every time samplemust be computed in paralell in order to find the Fourier coefficients, W , R, W and R arebidimensionnal matrices of size 2N + 1 by the number of conservative variables within eachcell. The dimension of size 2N + 1 contains the solution at each time sample for W and Ror the Fourier coefficients for W and R.

Input parameters specifications

Input file creation

NSCODE initialisation

Topology creation

Steady Flow Solver

Initialisation of the Unsteady Solver

Solve Flow for a time step

record outputs for the time step

Time Completed? no

yes

Time = Time+ ∆t

Post Processing

Figure 3.22 Flow chart of unsteady NSCODE Dual Time Stepping execution

To initialise NLFD simulations involving moving geometries, the same procedure as for thesteady solver is applied to each time sample, which is giving the far field boundary valuesto each cells. The moving meshes will provide different solutions at each time samples andthe higher modes of the solution will appear naturally. On non-forced unsteady simulations

43

however, as the geometry is not moving, higher modes may not appear since each time samplesare identical. Some strategies are thus required to obtain convergence of the solution on staticgeometries. For the purpose of this work, the strategy used is to impose a movement (pitchingor plundging) to the geometry for a few multigrid cycles before starting the computation ofthe static case. This allows all the different modes to appear as the simulation starts.

In the current framework, the FFTW (Fastest Fourier Transform in the West) library (Frigoand Johnson, 1998) is used to compute the direct and inverse Fourier transforms. Thelibrary, developed at the Massachusetts Institute of Technology, is optimised to computeFourier transforms with minimal computing costs. Using a third party optimised library forthis step thus saves development and simulation time.

W W R R

ikW

+ R∗

Figure 3.23 Flow chart of NLFD iteration algorithm on the state variables

3.4.3 Arbitrary Lagrangian Eulerian formulation

Many unsteady flow problems involve moving geometries. Using rigid grid movement, amodel must be used to account for the motion of the grid relative to the fluid. The principleof the Arbitrary Lagrangian Eulerian (ALE) formulation is to add to the convective fluxesthe part that is due to the sweeping of the cell faces through the fluid. Since this work onlycovers test cases with rigid grid movement, the Geometric Conservation Law (GCL), allowingarbitrary grid movement, is not verified. Modifying the formulation of the convective flux inequation 2.3

~FMC = ~FC − Vt ~W (3.1)

Where Vt is the mesh contra-variant velocity, in 2D

Vt = nx∂x

∂t+ ny

∂y

∂t(3.2)

In practice, this formulation means that the difference between the flow velocity and meshvelocity must be used to compute the convective flux vector. Along with this change, onemust also make sure that the rest of the numerical schemes remains consistent with the

44

modification of the flux vector. The velocities used in convective and absolute flux Jacobianand their eigenvalues must thus also use the difference between the fluid and mesh velocities.The treatment of the boundary conditions make no exceptions and must also use this velocitydifference.

3.4.4 Validation and Verification

Pitching NACA0012

The first test case is a pitching NACA0012 airfoil under Euler conditions. The freestreamMach number is 0.755. The sinusoidal pitching oscillation mean angle of attack (α0) is 0.016and the oscillation amplitude (αA) is 2.51 with a reduced frequency (k) of 0.0814. The centerof rotation is located at the quarter chord point of the airfoil. The mesh is reused from theprevious NACA0012 Euler test case from section 3.3.4. For the dual time-stepping solver, themesh displacement and velocity is calculated analytically at each time step. For the NLFDfunction, the mesh velocity and position are calculated analytically at each time sample andstored as such.

The dual time-stepping simulation was made using 150 multigrid cycles of steady simulationfollowed by 2 period of coarse time resolution to compute the transients without spendingtoo much time. The final time resolution was made using 100 time steps per period. Thedensity residual is reduced by ten orders of magnitude at each time step.

Figure 3.24 Euler pitching NACA0012 convergence characteristics versus multigrid cycles fordifferent number of modes

The NLFD simulation was made using 1 through 4 modes. This test case will be used to verifythe claims of the NLFD method to possess the same convergence rate in terms of iterations as

45

Figure 3.25 Euler pitching NACA0012 convergence characteristics versus CPU time for dif-ferent number of modes

a steady resolution. It will also verify that the computing cost of the solutions grows linearlywith a multiple of 2N+1. Table 3.9 shows the time to converge to machine accuracy, extractedfrom figure 3.25, for each number of modes used for the NLFD simulations. From figure 3.24,it is concluded that convergence to machine level is obtained after about 500 iterations. Ascan be observed, the NLFD computational cost scales as expected when varying the numberof modes of the Fourier series. The CPU time for an equivalent steady computation withthe same flow solver settings (CFL number, multi-grid levels, dissipation scheme) is addedas reference. The objective CPU time column represents the time each NLFD simulation isexpected to take if the direct and inverse Fourier transform are performed instantanously. Itis obtained by multiplying the steady resolution time by the factor of 2N + 1. The differencebetween the actual CPU time and the Theorical CPU time show that the Fourier transformstake about 33% of the total CPU time.

Table 3.9 CPU time to achieve 500 multigrid cycles for different number of modes used inthe NLFD resolution for the Euler pitching NACA0012 case

Number of modes CPU time(s) Objective CPU time(s) FFT weight in simulation timeSteady resolution 257 - -1mode 1149 771 32.9%2modes 1917 1285 33.0%3modes 2725 1799 34.0%4modes 3580 2313 35.4%

This test case is also used to test the chimera capabilities of the software with unsteadysimulations. Simulations of the test case using an overset mesh are completed with both

46

technique, DTS and NLFD method. This test case is ideal to test the chimera capability asit uses moving grids, the chimera preprocessor is thus called several times throughout theresolution. This means the solver must be able to communicate with the preprocessor toupdate the chimera interpolation informations for each time step or time sample dependingon the time discretisation method used. The overset mesh used is the same as the one usedpreviously in section 3.2.2 and illustraded in figure 3.2. The overset mesh simulation is doneusing two modes for the NLFD resolution.

Figure 3.26 shows the lift and drag coefficient hysteresis versus the instantaneous angle ofattack of the airfoil. Good agreement between the time accurate DTS solver, the NLFDsolver, the PMB software solution(Da Ronch et al., 2013) and experimental data (Landon).The chimera results also show very good agreement with the other results obtained using theone-to-one grid. The NLFD method yields good results when using at least two modes forthe resolution. Indeed, since the evolution of the drag coefficient during the period has twomaxima and minima, a single mode is not sufficient to describe such a shape, which wouldlead to inconsistent drag prediction.

Figure 3.26 Lift and drag hysteresis of the lift and drag coefficients versus the instantanousangle of attack

The time accuracy order of the DTS scheme in NSCODE is also assessed using this test case.The integral of the L2 norm of CL and CD are recorded in the steady periodic flow state toassess the accuracy orders. These parameters are computed as follows

‖CL‖ =n∑i=1

CL(ti)2∆t (3.3a)

‖CD‖ =n∑i=1

CD(ti)2∆t (3.3b)

47

Where n is the number of time steps over a period(T ), ∆t is the time step and CL(ti) andCD(ti) are respectively the value of the lift and drag coefficient obtained at the timestep i.The test case is computed using different values of time steps varying from 10 time steps perperiod up to 160 time steps per period.

As can be observed in table 3.10, both lift and drag norms achieve second order convergencewith respect to time resolution. This is expected as a second order backward differencescheme is used in the solver to compute the time derivatives. The convergence of the erroron the L2 norms of each coefficient is illustrated in figure 3.27.

Table 3.10 Convergence order and continuum estimates for pitching NACA0012 Dual Time-Stepping solutions

Time steps (∆t/T ) ‖CL‖ ‖CD‖per period

10 0.1 0.2472891128 0.00910416720 0.05 0.2550601978 0.009341509640 0.025 0.257244761 0.009399999480 0.0125 0.2577983417 0.0094117931160 0.00625 0.2579377802 0.009414387320 0.003125 0,257972786 0,009414983Continuum 0,257984521 0,009415161Order p 1,993945954 2,121784556

−2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1−7

−6

−5

−4

−3

−210

2040

80160

320

1020

4080

160320

log(

1∆t/T

)

log|φ−φ

∗ |

φ = ‖CL‖φ = ‖CD‖

Figure 3.27 Order of time convergence of NSCODE using the L2 norm integral of lift anddrag coefficient over a period for a pitching NACA0012 airfoil

48

Cylinder

The second test case is the observation of Von Karman vortices in the wake of a stationarycircular cylinder in laminar flow. The free stream Mach number is 0.3 and the Reynoldsnumber is 100. The objective of this test case is to show good agreement between the dualtime-stepping, the NLFD method and the literature. Figure 3.28 shows the mesh used forthe test case, which is a "O" mesh geometry containing 10000 cells with a wall grid spacingof 10−4 chord obtained from NASA’s CFL3D website1.

Using the DTS solver, the transient part of the flow is accelerated using asymmetric initiali-sation. This is done by computing the problem with the steady solver at a 45 angle-of-attackand then switching back to a zero angle of attack for unsteady resolution. Without an asym-metric initialisation, the symmetric grid would lead to symmetric vortices, which are knownto be non-physical, until machine errors would introduce small asymmetries in the flow thatwould grow into vortex shedding. For each time step, the solution is advanced in pseudo timeto decrease the density residual by five orders of magnitude. To compute the transient partbetween initialisation and steady periodic flow, three unsteady loops have been used with in-creasing time resolution. Each initialisation loop lasts four periods, and contain respectively5, 20 and 40 timesteps per period. The period used is based on the Strouhal number foundin the literature (Mosahebi and Nadarajah, 2013). The final resolution loop for the periodicflow contains 100 points per period. The explicit Runge-Kutta space resolution results arecompared to the ones obtained with the LU-SGS scheme.

Table 3.11 Strouhal numbers obtained for the cylinder under laminar flow conditions

StMosahebi and Nadarajah (2013) 0.16704Explicit RK DTS 0.17458LU-SGS DTS 0.17458Experimental results (Williamson, 1989) 0.16434

Using the NLFD method, the solution was initialized with 100 multigrid cycles where thecylinder oscillates in a plunging motion. Without this process, all time samples would haveidentical convergence and only the 0th mode would be non zero, one would thus need to waitfor machine errors to induce some non-zero values in the other modes of the solution.

Since the NLFD method requires to know in advance the exact Strouhal number in order toachieve machine accuracy, the Strouhal number obtained with the DTS simulation is usedto compute the NLFD simulation. However, the Strouhal number obtained from the DTS

1http://cfl3d.larc.nasa.goc/Cfl3dv6/cfl3dv6_testcases.html

http://cfl3d.larc.nasa.goc/Cfl3dv6/cfl3dv6_testcases.html

49

Figure 3.28 Mesh used for the circular cylinder simulations

Figure 3.29 Vorticity contours on the circular cylinder simulations

solver may not perfectly match the NLFD resolution. Analogously to a steady solution thatdoesn’t reach convergence on an unsteady test case, a wrong Strouhal number won’t allowthe unsteady terms to balance perfectly the convective and viscous terms of the residual.Therefore, density residual convergence only reached four orders of magnitudes. Table 3.11compares the Strouhal numbers from the literature, experimental results and NSCODE’sdual time stepping. The simulation was made using 4 modes for the Fourier series.

Multigrid technique was not used for this test case neither for the dual time stepping norfor the NLFD simulations. The reason being that the bigger cells and the use of first orderartificial dissipation on coarse meshes lead to steady solutions of the flow on coarse grids,changing the physics of the problem. The change in physics between fine and coarse grids leadthe multigrid technique to transfer wrong corrections to the fine grid, altering the solution.This situation is comparable to supersonic flows where multigrid tends to affect the solution.

Figure 3.30 shows the agreement between the implemented unsteady solvers. Each simulationyielded identical curves for lift and drag coefficients variation along a period. Figure 3.29shows the vorticity field behind a circular cylinder. It is observed that the vortices dissipatedafter approximately four chords lengths. This is mainly due to the rapidly expanding cell

50

Figure 3.30 Lift and drag coefficient variations over a period

sizes in the wake which tend to dissipate vortices. To study the dissipation of vortices witha finer grid, a simulation using overset grids containing a refined background was done. Theforeground mesh is a cropped version of the circular cylinder mesh used previously so thatit only extends three diameters length outside the cylinder. The background mesh of size256x256 contains rectangular cells, extends 50 diameters length in every directions from thecylinder and is refined near the center to provide better resolution of the vortices. Theassembly of the overset grids can be seen on figure 3.31. Using these grids, vortices wereobserved up to twelve cylinder lengths away from the cylinder as can be observed in figure3.32. This is an example of how overset grids can improve the quality of simulations.

A limitation in the usage of the NLFD method has also been encountered when simulatingthe flow around the circular cylinder. When increasing the number of modes used in thesimulation, the stability of the solver seems to decrease. Figure 3.33 compares the convergencecurves of the four modes resolution presented in the test case and an equivalent ten modesresolution. It is observed that while all modes used in the resolution converge when usingfour modes, most modes of the ten modes solution diverge.

51

Figure 3.31 Overset mesh used on the circular cylinder simulation

Figure 3.32 Vorticity contours on the circular cylinder simulation using an overset mesh

52

Figure 3.33 Convergence of each harmonic mode on a 4 modes (top) and 10 modes (bottom)NLFD resolution

53

CHAPTER 4 NUMERICAL RESULTS

This chapter applies the methods described earlier to application cases. Two particular casesare chosen to make best use of the methods:

• A high-lift airfoil optimisation;

• An unsteady simulation of turbulent flows in the frequency domain of a pitching airfoil.

The high-lift airfoil optimisation seeks to maximise the lift coefficient of the McDonnel Dou-glas Research airfoil by changing the flap’s position using a two dimensional approach anda three dimensional approach with infinite swept wing hypothesis. The second case soughtto introduce a turbulence component to the NLFD implementation in NSCODE. As a stepbefore using one and two equations turbulence models, an algebraic turbulence model is used.

4.1 High-lift airfoil optimisation

The aim here is to optimise the high-lift MDA geometry for a landing configuration. To doso, the highest maximum lift coefficient is sought to allow for lower approach speeds. Theoptimisation uses a full mapping of the maximum lift coefficient of the geometry when movingthe position of the flap in x and y direction. While requiring more computational resources,this method allows to capture the overall behaviour of the design space with respect to flapdisplacements. This problem was inspired by the work of Mavriplis and Mani.

The chimera method is used extensively in this study to modify the gap and overlap of theflap. In high-lift device vocabulary, the flap position variation in the x direction correspondsto a variation in the overlap while the variation in y direction corresponds to the gap be-tween the main element and the flap. Figure 4.1 illustrates the concepts of gap and overlapbetween the flap and the main element, note that in this study, x and y variations correspondrespectively to overlap and gap variation from the baseline MDA30N30P geometry. Sincethe mesh associated with each element is independent, no remeshing is needed to modifythe geometry. Only the flap mesh has to be rigidly moved to the new flap position and thechimera preprocessing has to be redone to account for the new cells overlap. The chimerapreprocessing can however be problematic in the tight gap between the flap and the mainairfoil since the chimera boundary requires a depth of a few cells in order to ensure an accu-rate communication between the meshes. A large buffer, for example of three or more cells,could end up computing the flow on cells using next to or contained inside solid boundaries

54

which would yield inaccurate results. Interpolation errors also arise as the interpolated cellsnear the chimera boundaries will likely obtain their data from much smaller near wall cellsof another mesh.

Figure 4.1 Gap and overlap definition for a flap

As a criteria, the smallest gap deemed acceptable for this analysis is determined as the small-est gap that allows two interpolated cells between solid wall boundaries and the computedregion of a mesh. This limits the maximum y deflection of the flap to 0.01% of the chordlength.

4.1.1 2.5D infinite swept wing

To show the importance of crossflow in high-lift aerodynamic design, the optimisation isconducted with unswept 2D and 30 swept wing conditions using the infinite-swept wingapproach as developed by Ghasemi et al.. The two CLmax maps obtained are compared inorder to assess the impact of sweep on high-lift design.

Figure 4.2 shows the difference in the CL–α curve between the 2D resolution of the MDAairfoil without sweep and a 30 sweep infinite swept wing simulation. As expected, the CLαratio between the 30 swept wing and the purely 2D wing is almost cos(30 ) between 4 and8.1 angle of attack and would be of exactly cos(30 ) in inviscid flow. The stall is also moreabrupt and occurs at a smaller attack angle and lift coefficient in the swept wing simulation.These differences show that a 2D high-lift analysis and optimization without considering theeffect of sweep leads to unrealistic multi-element designs.

To further illustrate the difference between a purely 2D flow and a 30 swept wing flow, figure4.3 show the difference in flowfields at 21 angle of attack. As is showed by the abrupt stallcharacteristics of the infinite swept wing flow in figure 4.2, the main element and the flap arecompletely stalled at 21 angle of attack. The purely 2D resolution however show a flowfieldwhere only a small portion of the trailing edge of the flap is stalled.

55

Figure 4.2 Cl–α curve of the MDA airfoil for a purely 2D simulation (0 sweep) and an infiniteswept wing simulation with a sweep of 30

Figure 4.3 Streamlines and pressure field over a MDA geometry for a 2D (or infinite 0 sweptwing) on the left and an infinite 30 swept wing on the right

56

4.1.2 Optimisation results

Figure 4.4 shows the results of the mapping for the 2D CLmax optimisation using purely 2Dsimulations on the left and 2.5D infinite swept wing approach on the right. As expected, theoptimum point was obtained with the same parameters combinations from both mappingand the CLmax obtained was much lower with the infinite swept wing approach.

The study also shows that swept wings would be much more sensitive to parameters change.A notable effect captured by the 2.5D approach that is not observed in the purely 2D approachis the loss of performance at a translation of 0.015 or 1.5% of the flap in the x direction. Thisposition roughly corresponds to the point were the flap no longer overlaps the main element.

Figure 4.4 Lift coefficient obtained for each x and y displacement of the flap in a 2D flow onthe left and an infinite 30 swept wing on the right

4.2 Turbulent NLFD case

4.2.1 Turbulence Model

The Baldwin Lomax turbulence model has been developped by Baldwin and Lomax (1978)to create an algebraic model for turbulence. The main reason leading to the choice of theBaldwin Lomax model here is to obtain turbulent results at low computational costs withoutextensive software changes. Since turbulence is currently computed decoupled from theconservative variables in the NSCODE framework, calculating the time derivative termsin turbulence models involving transport equation would require additional computationaleffort to recompute the Fourier transform of the conservative variables. The Baldwin Lomaxturbulence model being algebraic, it does not require to solve transport equation or theFourier coefficient of the conservative variables. It can directly be applied to the solution ateach time sample of the NLFD solution without changing the NLFD solver in other ways.

57

4.2.2 Pitching NACA64A010 airfoil

The test case CT6 from Landon is chosen to test the turbulence model usage with NLFDsimulations. The freestream Mach number is 0.796 and Reynolds number is 12.56x106.As mentionned previously, the Baldwin-Lomax turbulence model is used. The airfoil is aNACA64A010 profile undergoing a sinusoidal pitching motion around its quarter chord. Themean angle of attack of the motion is 0.00 with an amplitude of ±1.01 . The reducedfrequency kc is 0.202.

Figure 4.5 shows the "C" topology mesh used. The mesh has a size of 193x49, expands20 chords from the domain and its wall distance has a y+ of 1 or smaller on all the solidboundaries. The mesh parameters were selected to reproduce the study of Jameson et al.(2002). The mesh is again obtained with the NSGRID grid generator (Hasanzadeh et al.,2013).

Figure 4.5 Mesh used for the turbulent pitching NACA64A010 test case

Figure 4.6 shows the lift and moment coefficient hysteresis versus the angle of attack ofthe airfoil. Good agreement is observed between the different numerical methods and theliterature (Jameson et al., 2002). Lift hysteresis is also in accordance to experimental results.As few as 2 NLFD modes are required to correctly evaluate the lift.

Figure 4.7 and 4.8 show a snapshot of the solution at 0.65 while the airfoil is moving in anose down motion. The Mach contour and Cp distribution results are plotted using a NLFDresolution with 5 modes.

58

Figure 4.6 Lift and moment coefficients versus the angle of attack for the pitching turbulentNACA64A010 airfoil

59

Figure 4.7 Mach contours at 0.65 in a nose down motion of the turbulent NACA64A010airfoil

Figure 4.8 Cp distribution at 0.65 in a nose down motion of the turbulent NACA64A010airfoil

60

CHAPTER 5 CONCLUSION

5.1 Synthesis of Work

This work presented in details the developments of a steady explicit RANS solver to incor-porate various steady and unsteady solver schemes. Three main avenues have been exploredto adress the current challenges of CFD.

Firstly, the ability of the software to handle complex geometries were improved using multi-block and overset grid technique. Not only did overset grids proved to be adequate to handlehigh-lift airfoils, but they were also used to improve the resolution of the wake of a circularcyclinder. Indeed, vortices dissipation was extended from four to twelve diameter using acartesian background mesh refined in the wake region instead of a standard single block "O"type grid.

Secondly, a point implicit Jacobi preconditionner and an implicit LU-SGS scheme were im-plemented to improve the steady flow solver. Validation of these methods through varioustest cases showed the effectiveness of these developments. The Point Jacobi preconditionnerproved to be inefficient for Euler cases but very good on RANS cases that had more stretchedgrids. The implicit LU-SGS scheme proved to be generally more robust in terms of conver-gence rate per solver iteration. However, the additionnal computational costs for each solveriteration generally balances the gains in convergence rates. The method is thus more robustat a comparable computationnal cost to explicit and preconditionned point implicit schemes.Every schemes were verified and validated with other CFD solvers and second order spaceaccuracy was shown using a standard NACA0012 airfoil under inviscid flow conditions.

Thirdly, a Non Linear Frequency Domain solver was implemented in addition to the DualTime-Stepping scheme to efficiently solve periodic unsteady flows. For the case tested, theNLFD method proved to yield the same results as the DTS scheme for periodic flows usingsomehow few harmonic modes in the simulation. The number of solver iterations to convergea NLFD simulation was shown to be similar to a steady simulation and the computationnalcosts for each iteration scales linearly with the number of harmonic modes used. Theseproperties of the NLFD solver make it much faster to obtain periodic solutions than a DTSsimulation. The DTS time discretisation was verified to be of second order accuracy.

All these implementations were made with NSCODE using a carefully designed frameworkarchitecture. Low level laguage (C ) is used for computationaly intensive calculations, whilehigh-level language (Python) is used to control the various modules. The framework resides

61

inside a revision control system (Mercurial) to support the developpment of the code.

Using these developpments, two applications were performed

• Flap position optimisation to obtain the best possible CLmax;

• Study of a turbulent NACA64A010 test case;

The flap optimisation study took advantage of the ability of the code to handle complexgeometry and the improvements in steady solver to improve solution robustness. The studyshowed the optimal overlap position of the flap and the little influence of the gap on theCLmax. The benefit of using the infinite swept wing assumption was demonstrated. Theturbulent pitching airfoil case showed the potential of using turbulence models coupled withthe NLFD solver for efficient calculations of periodic flows.

5.2 Limitations of the Proposed Solution

Few limitations were encountered during software developments. First, the incompatibilityof the chimera technique with multigrid acceleration technique is an important downsideto the use of overset meshes. Indeed, as the multigrid method is one of the most effectivetechniques to obtain faster convergence of the flow solver, simulations using overset meshesare expected to be less performant.

A second limitation related to the use of overset grids is encountered when simulating mul-tiple body geometries with narrow gaps between them. Using overset meshes, the requiredminimum rows of cells to ensure connections between grids of different bodies may not berespected, degrading the accuracy of the solution. Although Soucy and Nadarajah (2009)showed a technique to obtain residual convergence to machine accuracy using the multigridtechnique on overset grids, the problem of coarse boundary definition on the coarse gridsremains unsolved when dealing with multiple bodies close to one another.

Another limitation was observed on the NLFD implementation as the number of harmonicmodes were increased, which is that a high number of modes in the computation tends to leadto slower convergence for each solver iterations and eventually divergence of the solution forvery large number of modes. A possible solution has been proposed by McMullen (2003) withthe Coarse grid Spectral Viscosity to adress this issue. Mundis and Mavriplis also recentlyshowed promising results with using up to 47 modes with the time spectral method to solvean Euler problem.

Benchmark runs showed that NSCODE parallelised portion stands at 94.6% using Amdahl’s

62

law, the maximum theoretical speedup is thus calculated at 18.5x. The parallel implementa-tion was made withOpenMP, limiting the execution of the code on shared memory computers.

5.3 Future Work

In terms of complex geometry handling, patch grid treatment is currently investigated tohandle overset grids on bodies intersecting each other. Patch grids are especially useful forwing-aileron configurations where there is no gap between the two bodies even though theyare moving relative to one another.

Full implicit solvers, such as GMRES scheme, are currently considered to be implementedto further increase the convergence rate of NSCODE. Compatibility between implementedNLFD solver and LU-SGS scheme are also considered.

Several improvements in the NLFD method are possible in the near future. First, the Gradi-ent Based Variable Time Period method (McMullen et al., 2002) would make a fine additionto the NLFD solver in cases where the Strouhal number of the phenomenon is not knownbeforehand. This method enables the NLFD solver to adjust the Strouhal number of thesimulation at each iteration of the solver in order to achieve machine accuracy when simu-lating unsteady periodic flows without forced motion. Another very interesting improvementcurrently investigated for the NLFD solver is to solve the transport equations of the conser-vative variables and of the turbulence models in a coupled manner. This modification wouldallow to decompose the turbulence models in the same fashion as the conservative variablesand produce fully turbulent simulations using the NLFD solver and turbulence models suchas Spalart-Allmaras or k − ω.

Finally, the algorithms presented can all be used in aero-elastic problems, as well as extendedto 3D flows.

63

REFERENCES

ALLMARAS, S. R., “Analysis of a local matrix preconditioner for the 2-d navier-stokesequations,” in 11th Computational FLuid Dynamics Conference. Orlando, FL: AIAA,1993.

AREVALO, S., ATWOOD, C., BELL, P., BLACKER, T., DEY, S., FISHER, D., FISHER,D., GENALIS, P., GORSKI, J., HARRIS, A. et al., “A new dod initiative: the computa-tional research and engineering acquisition tools and environments (create) program,” inJournal of Physics: Conference Series, vol. 125, no. 1. IOP Publishing, 2008, p. 012090.

BALDWIN, B. S. and LOMAX, H., Thin layer approximation and algebraic model forseparated turbulent flows. American Institute of Aeronautics and Astronautics, 1978, vol.257.

BLAZEK, J., Computational Fluid Dynamics: Principles and Applications:(Book with ac-companying CD). Elsevier, 2005.

CAGNONE, J., SERMEUS, K., NADARAJAH, S. K., and LAURENDEAU, E., “Implicitmultigrid schemes for challenging aerodynamic simulations on block-structured grids,” Com-puters & Fluids, vol. 44, no. 1, pp. 314–327, 2011.

DA RONCH, A., MCCRACKEN, A., BADCOCK, K., WIDHALM, M., and CAM-POBASSO, M., “Linear frequency domain and harmonic balance predictions of dynamicderivatives,” Journal of Aircraft, vol. 50, no. 3, pp. 694–707, 2013.

DELOZE, T. and LAURENDEAU, E., “Nsmb contribution to the 2nd high lift predictionworkshop,” in 52nd Aerospace Sciences Meeting, aIAA Paper 2014–0913.

FRIGO, M. and JOHNSON, S. G., “Fftw: An adaptive software architecture for the fft,” inAcoustics, Speech and Signal Processing, 1998. Proceedings of the 1998 IEEE InternationalConference on, vol. 3. IEEE, 1998, pp. 1381–1384.

GHASEMI, S., MOSAHEBI, A., and LAURENDEAU, E., “A two-dimensional/infiniteswept wing navier-stokes solver,” in 52nd Aerospace Sciences Meeting, aIAA Paper 2014–0557.

HALL, K. C., THOMAS, J. P., and CLARK, W. S., “Computation of Unsteady NonlinearFlows in Cascades Using Harmonic Balance Technique,” AIAA Journal, vol. 40, no. 5, pp.879–886, May 2002.

64

HASANZADEH, K., MOSAHEBI, A., LAURENDEAU, E., and PARASCHIVOIU, I., “Val-idation and verification of multi-steps icing calculation using canice2d-ns code,” in AIAAFluid Dynamics and Co-located Conferences and Exhibit. San Diego, California, USA:AIAA, 2013.

HIRSCH, C., Numerical Computation of Internal and External Flows: The Fundamentalsof Computational Fluid Dynamics: The Fundamentals of Computational Fluid Dynamics.Butterworth-Heinemann, 2007, vol. 1.

JAMESON, A., “Time Dependent Calculations Using Multigrid, with Applications to Un-steady Flows Past Airfoils and Wings,” in AIAA 10th Computational Fluid Dynamics Con-ference, vol. AIAA 91-1596, Honolulu, HI, 1991.

JAMESON, A., SCHMIDT, W., and TURKEL, E., “Numerical solution of the euler equa-tions by finite volume methods using runge-kutta time-stepping schemes,” AIAA paper, vol.1259, 1981.

JAMESON, A., “Solution of the euler equations for two dimensional transonic flow by amultigrid method,” Applied mathematics and computation, vol. 13, no. 3, pp. 327–355, 1983.

JAMESON, A., “Multigrid algorithms for compressible flow calculations,” in MultigridMethods II. Springer, 1986, pp. 166–201.

JAMESON, A. and BAKER, T. J., “Solution of the euler equations for complex configura-tions,” AIAA paper, vol. 83, 1983.

JAMESON, A. and YOON, S., “Lower-upper implicit schemes with multiple grids for theeuler equations,” vol. 25, no. 7, 1987, pp. 929–935.

JAMESON, A., ALONSO, J., and MCMULLEN, M., “Application of a non-linear frequencydomain solver to the euler and navier–stokes equations,” AIAA paper, vol. 13625, 2002.

LANDON, R., “Naca 0012 oscillating and transient pitching, data set 3,” Compendium ofunsteady aerodynamic measurement, AGARD.

LEVESQUE, A. T., PIGEON, A., DELOZE, T., and LAURENDEAU, E., “An overset grid2d/infinite swept wing urans solver using recursive cartesian virtual grid method,” in 53RDAIAA AEROSPACE SCIENCES MEETING, aIAA Paper 2015–0912.

LIAO, W., CAI, J., and TSAI, H. M., “A multigrid overset grid flow solver with implicithole cutting method,” Computer Methods in Applied Mechanics and Engineering, vol. 196,no. 9–12, pp. 1701–1715, 2007.

65

MARTINELLI, L., “Calculation of viscous flows with a multigrid method,” Ph.D. disserta-tion, Dept. of Mechanical and Aerospace Engineering, Princeton University, NJ., 01 1987.

MAVRIPLIS, D. J. and MANI, K., “Adjoint-based shape optimisation of high-lift airfoilsusing the nsu2d unstructured mesh solver,” in 52nd Aerospace Sciences Meeting, aIAAPaper 2014–0554.

MCMULLEN, M., JAMESON, A., and ALONSO, J., “Acceleration of Convergence to aPeriodic Steady State in Turbomachinery Flows,” in 39th AIAA Aerospace Sciences Meeting& Exhibit, vol. AIAA 2001-0152, Reno, NV, 2001.

MCMULLEN, M., JAMESON, A., and ALONSO, J., “Application of a Non-Linear Fre-quency Domain Solver to the Euler and Navier-Stokes Equations,” in 40th AIAA AerospaceSciences Meeting & Exhibit, vol. AIAA 2002-0120, Reno, NV, 2002.

MCMULLEN, M. S., “The application of non-linear frequency domain methods to the eulerand navier-stokes equations,” Ph.D. dissertation, Citeseer, 2003.

MORINISHI, K., “A finite difference solution of the euler equations on non-body-fittedcartesian grids,” Computers & Fluids, vol. 21, no. 3, pp. 331–344, 1992.

MOSAHEBI, A. and NADARAJAH, S., “An adaptative Non-Linear Frequency Domainmethod for viscous flows,” Computers & Fluids, vol. 75, pp. 140–154, 2013.

MUNDIS, N. L. and MAVRIPLIS, D. J., “An efficient flexible gmres solver for the fully-coupled time-spectral aeroelastic system.”

PIERCE, N. and GILES, B., “Preconditioned multigrid methods for compressible flow cal-culations on streched meshes,” Journal of Computational Physics, vol. 136, no. CP975772,pp. 425–445, 1997.

PIGEON, A., “Accélération de la résolution des équations de type urans sur des ailesd’avions transsoniques,” Master’s thesis, Polytechnique Montreal, QC, CANADA, 2015.

PIGEON, A., LEVESQUE, A. T., and LAURENDEAU, E., “Two-dimensional navier-stokesflow solver developments at École polytechnique de montréal,” in CFD Society of Canada22 nd Annual Conference. Toronto, CA: CFDsc, 2014.

PULLIAM, T. H. and STEGER, J. L., “Recent improvements in efficiency, accuracy, andconvergence for implicit approximate factorization algorithms,” AIAA paper, vol. 854360,1985.

66

ROE, P., “Approximate riemann solvers, parameter vectors, and difference schemes,” Jour-nal of Computational Physics, vol. 43, pp. 357–372, 1981.

RUMSEY, C. L., SLOTNICK, J., LONG, M., STUEVER, R. A., and WAYMAN, T. R.,“Summary of the first aiaa cfd high-lift prediction workshop,” Journal of Aircraft, vol. 48,no. 6, pp. 2068–2079, 2011.

SHAROV, D. and NAKAHASHI, K., “Reordering of 3-d hybrid unstructured grids forvectorized lu-sgs navier-stokes computations,” AIAA paper, vol. 97, pp. 2102–2117, 1997.

SONI, K., CHANDAR, D. D. J., and SITARAMAN, J., “Development of an overset gridcomputational fluid dynamics solver on graphical processing units,” Computers & Fluids,vol. 58, no. 58, pp. 1–14, 2012.

SOUCY, O. and NADARAJAH, S. K., “A nlfd method for the simulation of periodic un-steady flows for overset meshes.” in 19th AIAA Computational Fluid Dynamics, 2009.

SOUTH JR, J. C. and BRANDT, A., “Application of a multi-level grid method to transonicflow calculations,” 1976.

SUHS, N. E., ROGERS, S. E., DIETZ, W. E., and KWAK, D., “Pegasus 5: An automatedpre-processor for overset-grid cfd,” 2002.

SWANSON, R. and TURKEL, E., “On some numerical dissipation schemes,” Journal ofComputational Physics, vol. 147, no. CP986100, pp. 518–544, 1998.

SWANSON, R. R. and TURKEL, E., “Multistage schemes with multigrid for euler andnavier-stokes equations,” NASA Technical Paper, no. 3631, 1997.

THIBERT, J. J., “Cfd platforms: an introduction to onera’s softwares.”

VASSBERG, J. C. and JAMESON, A., “In pursuit of grid convergence for two-dimensionaleuler solutions,” AIAA Paper, no. 81-1259, pp. 1152–1166, 1981.

VERSTEEG, H. K. and MALALASEKERA, W., An introduction to computational fluiddynamics: the finite volume method. Pearson Education, 2007.

WANG, Z. J. and PARTHASARATHY, V., “A fully automated chimera methodology formultiple moving body problems,” International Journal for Numerical Methods in Fluids,vol. 33, no. 7, pp. 919–938, 2000.

67

WEBER, C., “Developpement de methodes implicites pour les equations de navier stokesmoyennees et la simulation des grandes echelles: Application al aerodynamique externe,”These de doctorat, INPT Toulouse, 1998.

WILLIAMSON, C., “Oblique and parallel modes of vortex shedding in the wake of a circularcylinder at low reynolds numbers,” Journal of Fluid Mechanics, vol. 206, pp. 579–627, 1989.

WRIGHT, M. J., CANDLER, G. V., and PRAMPOLINI, M., “Data-parallel lower-upperrelaxation method for the navier-stokes equations,” AIAA journal, vol. 34, no. 7, pp. 1371–1377, 1996.

YOON, S. and JAMESON, A., An Multigrid LU-SSOR Scheme for Approximate NewtonIteration Applied to the Euler Equations. National Aeronautics and Space Administration,1986.

YOON, S. and JAMESON, A., “Lower-upper symmetric-gauss-seidel method for the eulerand navier-stokes equations,” AIAA journal, vol. 26, no. 9, pp. 1025–1026, 1988.

68

APPENDIX A TOPOLOGY FILE EXAMPLE

This annex contains the structure of a topology input file of NSCODE. It is an example fora mesh around a NACA0012 airfoil. The mesh was initially a single block "O" mesh dividedinto four equal blocks. The mesh size was initially 65 by 65 nodes.

NACA0012417 6517 6517 6517 65BC is ie js je icomp

block# 1 nsubface 4CON 1 1 1 65 0

4 17 17 1 65 0CON 17 17 1 65 0

2 1 1 1 65 0WAL 1 17 1 1 1FAR 1 17 65 65 0


1 17 17 1 65 0CON 17 17 1 65 0

3 1 1 1 65 0WAL 1 17 1 1 1FAR 1 17 65 65 0


2 17 17 1 65 0CON 17 17 1 65 0

4 1 1 1 65 0WAL 1 17 1 1 1FAR 1 17 65 65 0


3 17 17 1 65 0CON 17 17 1 65 0

1 1 1 1 65 0WAL 1 17 1 1 1FAR 1 17 65 65 0

69

APPENDIX B VISCOUS JACOBIANS

The current Appendix formulates the different viscous flux Jacobians used throughout theCFD community. For simplification purposes, only 2D Jacobians in the I computationnaldirection will be written.

The first formulation is from Blazek (2005) and uses the Thin Shear Layer approximation.

Av = µ

Ω

0 0 0 0b21 a1∂I (ρ−1) a2∂I (ρ−1) 0b31 a2∂I (ρ−1) a3∂I (ρ−1) 0b41 b42 b43 a4∂I (ρ−1)

(B.1)

where,

∂I (φ) = ∂φ

∂I

a1 = 43I

2x + I2

y

a2 = 13IxIy

a3 = I2x + 4

3I2y

a4 =(γ

Pr

) (I2x + I2

y

)b21 = −a1∂I

(u

ρ

)− a2∂I

(v

ρ

)

b31 = −a2∂I

(u

ρ

)− a3∂I

(v

ρ

)

b41 = −a4∂I

((u2 + v2)

ρ− E

ρ

)− a1∂I

(u2

ρ

)

−2a2∂I

(uv

ρ

)− a3∂I

(v2

ρ

)

b42 = −a4∂I

(u

ρ

)− b21

b43 = −a4∂I

(v

ρ

)− b31 (B.2)

70

The second formulation from Sharov and Nakahashi (1997) consists of taking a diagonalmatrix with the viscous spectral radius on the diagonal elements.

Av = µ

Ω

λIvisc 0 0 0

0 λIvisc 0 00 0 λIvisc 00 0 0 λIvisc

(B.3)

with λIvisc the viscous spectral radius in the I direction formulated as (Blazek, 2005)

λIvisc = max

( 43ρ,

γ

ρ

)(µLPrL

+ µTPrT

) (∆SI)2

Ω (B.4)

PrL and PrT are the Prandtl numbers using respectively the laminar and turbulent viscosity.

The third formulation comes from the European NSMB flow solver (Weber, 1998). Theviscous Jacobian is decomposed into its cartesian components

Bxv = ∂Fv

∂Ux, By

v = ∂Fv∂Uy

(B.5)

With U the primitive variables

Bxv =

0 0 0 00 4

3µsx µsy 00 −2

3µsy µsx 0−kTsx

ρ23µ (2usx − vsy) µ (vsx − usy) kTsx

p

(B.6)

Byv =

0 0 0 00 µsy −2

3µsx 00 µsx

43µsy 0

−kTsy

ρ µ (vsx − usy) 23µ (−usx + 2vsy) kTsy

p

(B.7)

The Jacobians must then be transformed to the conservative variables formulation beforebeing assembled

Aψv = Bψv ·

∂W

∂U(B.8)

Av = Ω(nxA

xv + nyA

yv

)(B.9)

With Ω being the surface of the cell face.

universitÉdemontrÉal ... · method to allow for even more ... a point implicit point-jacobi ......

Documents